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Determinants and Matrices A. C. Aitken, D.Sc., F.R.S. 

Statistical Mathematics A. C. Aitken, D.Sc., F.R.S. 

The Theory of Ordinary 

Differential Equations J. C. Burkill, Sc.D., F.R.S. 

Waves C. A. Coulson, D.Sc., F.R.S. 

Electricity C. A. Coulson, D.Sc., F.R.S. 

Protective Geometry T. E. Faulkner, Ph.D. 

Integration R. P. Gillespie, Ph.D. 

Partial Differentiation R. P. Gillespie, Ph.D. 

Infinite Series J. M. Hyslop, D.Sc. 

Integration of Ordinary Differential Equations E. L. Ince, D.Sc. 

Introduction to the Theory of 
Finite Groups ... ... ... ... W. Ledermann, Ph.D., D.Sc. 

German-English S. Macintyre, Ph.D., and E. Witte, M.A. 

Mathematical Vocabulary 

Analytical Geometry of 
Three Dimensions W. H. McCrea, Ph.D., F.R.S. 

Topology E. M. Patterson, Ph.D. 

Functions of a Complex Variable ... E. G. Phillips, M.A., M.Sc. 

Special Relativity W. Rindler, Ph.D. 

Volume and Integral ... W. W. Rogosinski, Dr.Phil., F.R.S. 

Vector Methods D. E. Rutherford, D.Sc., Dr.Math. 

Classical Mechanics D. E. Rutherford, D.Sc., Dr.Math. 

Fluid Dynamics D. E. Rutherford, D.Sc., Dr.Math. 

Special Functions of Mathematical 

Physics and Chemistry I. N. Sneddon, D.Sc. 

Tensor Calculus* B. Spain, Ph.D. 

Theory of Equations H. W. Turnbull, F.R.S. 




E. G. PHILLIPS, M.A., M.So. 




With 17 Figure* 


REPBINTED . . . 1961 



CHANGES that have been made in recent editions include 
a set of Miscellaneous Examples at the end of the book 
and an independent proof of Liouville's theorem has been 
given. In this edition, the proof of the Example on 
page 60 has been altered. 

Limitations of space made it necessary for me to confine 
myself to the more essential aspects of the theory and its 
applications, but I have aimed at including those parts of 
the subject which are most useful to Honours students. 
Many readers may desire to extend their knowledge of the 
subject beyond the limits of the present book. Such 
readers are recommended to study the standard treatises of 
Copson, Functions of a Complex Variable (Oxford, 1935), 
and Titchmarsh, Theory of Functions (Oxford, 1939). I 
take this opportunity of acknowledging my constant 
indebtedness to these works both in material and 

I have presupposed a knowledge of Real Variable 
Theory corresponding approximately to the content of 
my Course of Analysis (Cambridge, Second Edition, 
1939). References are occasionally given to this book in 
footnotes as P.A. 

I wish to express my thanks to all those friends who 
have made helpful suggestions. In particular, I mention 
two of my colleagues, Mr A. C. Stevenson, of University 
College, London, who read the proofs of the first edition, 
and Prof. H. Davenport, F.R.S., who very kindly suggested 
a number of improvements for the second edition. I desire 
also to express my gratitude to the publishers for the careful 
and efficient way in which they have carried out their 
duties. - E Q p 

BANGOR, October 1956 



PREFACE p. vii 



< Complex Numbers ....... I 

Sets of Points in the Argand Diagram ... 7 
Functions of a Complex Variable .... 9 

Regular Functions . . . . . . .11 

Conjugate Functions . . . . . .14 

^JPower Series 17 

The Elementary Functions . . . . .20 

Many -valued Functions . . . . . .26 

Examples I ........ 29 


Isogonal and Conformal Transformations . . .32 
Harmonic Functions . . . . . .37 

The Bilinear Transformation ..... 40 

Geometrical Inversion ...... 43 

The Critical Points 45 

Coaxal Circles 46 

Invariance of the Cross-Ratio 49 

Some special Mdbius' Transformations . . .51 
Examples II . , . . .54 




The Transformations 10 = 2". . . .68 
w = z* 60 

w = \/z 62 

w = tan 2 (frr\/z) 64 

Combinations of w == z* with Mobius' Transformations 66 
Exponential and Logarithmic Transformations . . 70 
Transformations involving Confocal Conies . . 71 

z = c sin w ........ 74 

Joukowski's Aerofoil . . . . . .76 

Tables of Important Transformations . . .78 
Schwarz-Christoffel Transformation .... 80 

Examples III ........ 81 


Complex Integration ...... 85 

Cauchy's Theorem . . . . . . .89 

The Derivatives of a Regular Function ... 93 

Taylor's Theorem 95 

Liouville's Theorem . . . . . .96 

Laurent's Theorem . . . . . . .97 

Zeros and Singularities ...... 98 

Rational Functions . . . . . . .103 

Analytic Continuation . . . . . .103 

' Poles and Zeros of Meromorphic Functions . .107 
Rouch6's Theorem . . . . . . .108 

The Maximum-Modulus Principle . . . .109 

Examples IV Ill 


The Residue Theorem ...... 

Integration round the Unit Circle 117 



Evaluation of Infinite Integrals . . . .119 

Jordan's Lemma . . . . . . .122 

Integrals involving Many -valued Functions . ,126 
Integrals deduced from Known Integrals . . .128 
Expansion of a Meromorphic Function . . .131 
Summation of Series . . . . . .133 

Examples V ........ 135 


INDEX 143 



1. Complex Numbers 

This book is concerned essentially with the application 
of the methods of the differential and integral calculus 
to complex numbers. A number of the form a+ij8, where 
t is V(~~ 1) an( l a an d j3 are real numbers, is called a 
complex number ; and, although complex numbers are 
capable of a geometrical interpretation, it is important 
to give a definition of them which depends only on real 
numbers. Complex numbers first became necessary in the 
study of algebraic equations. It is desirable to be able to 
say that every quadratic equation has two roots, every 
cubic equation has three roots, and so on. If real numbers 
only are considered, the equation a +l = has no roots 
and x 3 l = has only one. Every generalisation of 
number first presented itself as needed for some simple 
problem, but extensions of number are not created by 
the mere need of them ; they are created by the definition, 
and our object is now to define complex numbers. 

By choosing one of several possible lines of procedure, 
we define a complex number as an ordered pair of real 
numbers. Thus (4, 3), (\/2, e), (J, TT) are complex numbers. 
If we write z = (x, j/), x is called the real part, and y 
the imaginary part, of the complex number z. 

(i) Two complex numbers are equal if, and only if, 
their real and imaginary parts are separately equal. The 
equation z = z' implies that both x = x' and y = y'. 


(ii) The modulus of z, written | z |, is defined to be 
+ \/(x*-\-y*). H> follows immediately from the definition 
that | z | = if, and only if , x = and y = 0. 

(iii) The fundamental operations. 
If z = (z, y), z' = (#', y') we have the following definitions : 

(1) z+z' is (*+*', y+y'). 

(2) -zis(-*, -t/). 

(3) z-z' = z+(-z') is (z-a;', y-y'). 

(4) 22' is (xx'yy' 9 xy'+x'y). 

If the fundamental operations are thus defined, we easily 
see that the fundamental laws of algebra are all satisfied. 
(a) The commutative and associative laws of addition hold : 

(b) The same laws of multiplication hold : 

(o) TAe distributive law holds : 

As an example of the method, we show that the com- 
mutative law of multiplication holds. The* others are 
proved similarly. 

We have thus seen that complex numbers, as defined 
above, obey the fundamental laws of the algebra of real 
numbers : hence their algebra will be identical in /orm, 
though not in meaning, with the algebra of real numbers. 

We observe that there is no order among complex 
numbers. As applied to complex numbers, the phrases 
" greater than " or " less than " have no meaning. In- 
equalities can only occur in relations between the moduli 
of complex numbers. 


(iv) The definition of division. 

Consider the equation z = z', where z = (x, y), 
= (, i)) 9 z' = (x' 9 y') t then we have 

so that xyr\ = z' 

and, on solving for and 77, 

_ yy'+xx' _ xy'x'y 

provided that | z | ? 0. Hence, if | ^ | ^ 0, there is a 
unique solution, and = (f , TJ) is the quotient z'/z. 

Division by a complex number whose modulus is zero 
is meaningless ; this conforms with the algebra of real 
numbers, in which division by zero is meaningless. 

The abbreviated notation. 

It is customary to denote a complex number whose 
imaginary part is zero by the real-number symbol a?. If 
we adopt this practice, it is essential to realise that x may 
have two meanings (i) the real number x, and (ii) the 
complex number (x, 0). Although in theory it is important 
to distinguish between (i) and (ii), in practice it is legitimate 
to confuse them ; and if we use the abbreviated notation, 
in which x stands for (x, 0) and y for (y, 0), then 

x+y = (*, 0)+(y, 0) = (x+y, 0), 
xy = (x, 0) . (y, 0) = (x. y-0.0, x . 0+0 . y) = (xy, 0). 

Hence, so far as sums and products are concerned, complex 
numbers whose imaginary parts are zero can be treated 
as though they were real numbers. It is customary to 
denote the complex number (0, 1) by . With this 
convention, t 2 = (0, 1) . (0, 1) = ( 1, 0), so that i may 
be regarded as the square root of the real number 1. 
On using the abbreviated notation, it follows that 

(x, y) = x+iy, 


for, since % = (0, 1), we have 

= (*,0)+(0.y-1.0,0.0+l.y) 

= (a?, 0)+(0, y) = (a+0, 0+y) = (x, y). 

In virtue of this relation we see that, in any operation 
involving sums and products, it is allowable to treat 
x, y and i as though they were ordinary real numbers, 
with the proviso that i* must always be replaced by 1. 

2. Conjugate Complex Numbers 

If z = x+iy, it is customary to write x = Rz, y = Iz. 
The number x iy is said to be conjugate to z and is 
usually denoted by z. It readily follows that the numbers 
conjugate to z l -\-z 2 and 2 t 2 2 are Zi+z^ and 2^23 respectively. 

Proofs of theorems on complex numbers are often 
considerably simplified by the use of conjugate complex 
numbers, in virtue of the relations, easily proved, 

| Z | f == 22, 2R2 = Z+Z, 2ilz = 22. 

To prove that the modulus of the product of two complex 
numbers is the product of their moduli, we proceed as follows : 

2 = 

and so, since the modulus of a complex number is never 

I *i*a H I *i I I * ! 

Theorem. The modulus of the sum of two complex 
numbers cannot exceed the sum of their moduli. 

2 = 

and so 


a result which can be readily extended by induction to 
any finite number of complex numbers. 

In .a similar way we can prove another useful result, 


We have 


= (l*i H*i I) 2 ; 

\*i~**\ > 1(1 *i H * 1)1- 

3. Geometrical Representation of Complex 

If we denote (# 2 +y 2 )* by r, and choose so that 
r cos 6 = x, r sin = y, then r and are clearly the radius 

FIG. 1. 

vector and vectorial angle of the point P, (x, y), referred 
to an origin and rectangular axes Ox, Oy. It is clear 
that any complex number can be represented geometrically 
by the point P, whose Cartesian coordinates are (x, y) 
or whose polar coordinates are (r, 0), and the representation 
of complex numbers thus afforded is called the Argand 

By the definition already given, it is evident that r 
is the modulus of z = (x, y) ; the angle is called the 
argument of z, written = arg z. The argument is not 
unique, for if be a value of the argument, so also is 


2/17T+0, (n = 0, 1 2, ...) The principal value of arg z 
is that which satisfies the inequalities ?r<arg Z^TT. 

Let P! and P 2 (in fig. 1) be the points z l and z 2 , then 
we can represent addition in the following way. Through 
P l9 draw PjP 8 equal to, and parallel to QP 2 . Then P 8 
has coordinates (a^+^a* yi+^a) an d 8O -Ps represents the 
point z t +z 2 . 

In vectorial notation, 

QP 3 = 

Similarly, we have, if P 8 is the point 2 3 , 

It is convenient to write cis0 for cos^+fsin^. If 
z l = r l cis0 l9 z 2 = r 2 cis^ 2 , ..., z n = r n cis0 n , then, by 
de Moivre's theorem, 

which readily exhibits the fact that the modulus and 
argument of a product are equal respectively to the 
product of the moduli and the sum of the arguments of 
the factors. In particular, if n be a positive integer and 
z = r cis 0, z n = r n cis n$. 

^ = ^1 cis ^-fla). 

Z 2 r 2 

If n is a positive integer, there are n distinct values of 
z l i*. If m is any integer, since 

j cis 1 = cis0, 

\ / 

it follows that r*-t n cis{(0+2w7r)/n} is an nth root of z=r cis0. 
If we substitute the numbers 0, 1, 2, ... n 1 in succession 
for m, we obtain n distinct values of z l l n ; and the sub- 
stitution of other integers for m merely gives rise to 
repetitions of these values. Also, there can be no other 


values, since z l l n is a root of the equation u n = z which 
cannot have more than n roots. 

Similarly, if p and q are integers prime to each other 
and q is positive, 

where m = 0, 1, 2, ..., ql. 

By considering the modulus and argument of a complex 
number, the operation of multiplying any complex number 
x+iy by i is easily seen to be equivalent to turning the line 
OP through a right-angle in the positive (counter-clockwise) 
sense. We have just seen that 

(- 1 ) 

\z % / 

zj) = arg Zx+arg z 2> arg - = arg z l arg z a , 

\z % / 

so that the formal process of " taking arguments " is similar 
to that of " taking logarithms." Hence, if arg (x+iy) = a, 

arg i(x+iy) = arg i+a,Tg(x+iy) = JTT+OU 
Since | i \ = 1, multiplying by i leaves | x+iy \ unaltered. 

4. Sets of Points in the Argand Diagram 

We now explain some of the terminology necessary 
for dealing with sets of complex numbers in the Argand 
diagram. We shall use such terms as domain, contour, 
inside and outside of a closed contour, without more precise 
definition than geometrical intuition requires. The general 
study of such questions as the precise determination of 
the inside and outside of a closed contour is not so easy 
as our intuitions might lead us to expect.* For our 
present purpose, however, we shall find that no difficulties 
arise from our relying upon geometrical intuition. 

By a neighbourhood of a point z in the Argand 
diagram, we mean the set of all points z such that |z--z |<, 
where is a given positive number. A point z is said 

* For further information, see e.g. Dienee, The Taylor Seriet 
(Oxford, 1931), Ch. VI. 


to be a limit point of a set of points S 9 if every neighbour- 
hood of z contains a point of S other than z . The 
definition implies that every neighbourhood of a limit 
point z contains an infinite number of points of S. For, 
the neighbourhood | z z |< contains a point z 1 of S 
distinct from z , the neighbourhood | z z |<| z l z | 
contains a point z 2 of S distinct from z and so on 

The limit points of a set are not necessarily points of 
the set. If, however, every limit point of the set belongs 
to the set, we say that the set is closed. There are two 
types of limit points, interior points and boundary points. 
A limit point z of S is an interior point if there exists 
a neighbourhood of z which consists entirely of points 
of S. A limit point which is not an interior point is a 
boundary point. 

A set which consists entirely of interior points is said 
to be an open set. 

It should be observed that a set need not be eithet open 
or closed. An example of such a set is that consisting of the 
point z = 1 and all the points for which | z | <1. 

We now define a Jordan curve. 

The equation z = x(t)+iy(t), where x(t) and y(t) are 
real continuous functions of the real variable t, defined 
in the range a <J<j8, determines a set of points in the 
Argand diagram which is called a continuous arc. A 
point z x is a multiple point of the arc, if the equation 
z l = x(t) +iy(t) is satisfied by more than one value of t in 
the given range. 

A continuous arc without multiple points is called a 
Jordan arc. If the points corresponding to the values 
a and j3 coincide, the arc, which has only one multiple 
point, a double point corresponding to the terminal values 
a and j8 of t, is called a simple closed Jordan curve. 

A set of points is said to be bounded if there exists 
a constant K such that | z | ^.K is satisfied for all points 


z of the set. If no such number K exists the set is 

A domain is defined as follows: 

A set of points in the Argand diagram is said to be 
connex if every pair of its points can be joined by a 
polygonal arc which consists only of points of the set. 
An open domain is an open connex set of points. The 
set, obtained by adding to an open domain its boundary 
points, is called a closed domain. 

The Jordan curve theorem states that a simple closed 
Jordan curve divides the plane into two open domains which 
have the curve as common boundary. Of these domains 
one is bounded and it is called the interior, the other, 
which is unbounded, is called the exterior. Although 
the result stated seems quite obvious, the proof is very 
complicated and difficult. When using simple closed 
Jordan curves consisting of a few straight lines and circular 
arcs, geometrical intuition makes it obvious which is the 
interior and which is the exterior domain. 

For example, the circle | z \ = R divides the Argand 
diagram into two separated open domains | z \<R and 
| z | >7?. Of these the former is a bounded domain and is 
the interior of the circle | z \ = R ; the latter, which is 
unbounded, is the exterior of the circle | z \ = R. 

In complex variable theory we complete the complex 
plane by adding a single point at infinity. This point is 
defined to be the point corresponding to the origin by the 
transformation z' = 1/2. 

5. Functions of a Complex Variable. Continuity 

If u?(= u+iv) and z(= x-}-iy) are any two complex 
numbers, we might say that w is a function of z, w = /(z), if, 
to every value of z in a certain domain D, there correspond 
one or more values of w. This definition, similar to that 
given for real variables, is quite legitimate, but it is futile 
because it is too wide. On this definition, a function of 


the complex variable z is exactly the same thing as a 
complex function 

u(x, y)+iv(x,y) 

of two real variables x and y. 

For functions defined in this way, the definition of 
continuity is exactly the same as that for functions of a 
real variable. The function /(z) is continuous at the 
point Z Q if, given any e, >0, we can find a number 8 such that 

for all points z of D satisfying | z z 1 <8. The number 8 
depends on e and also, in general, upon z . If it is 
possible to find a number h(e) independent of z , such that 
\f( z )f( z o) |< holds for every pair of points z, z of the 
domain D for which |z z |<A, then /(z) is said to be 
uniformly continuous in D. It can be proved that a 
function which is continuous in a bounded closed domain 
is uniformly continuous there.* 

It is easy to show that this definition of continuity is 
equivalent to the statement that a continuous function 
of z is merely a continuous complex function of the two 
variables x and y, for, if 

/(z) = u(x, y)+iv(x, y), 

when /(z) is continuous on the above definition, so are 
u(x, y) and v(x y y) ; and conversely, if u and v are continuous 
functions of x and y, /(z) is a continuous function of z. 

The only class of functions of z which is of any practical 
utility is the class of functions to which the process of 
differentiation can be applied. 

6. Differentiability 

We next consider whether the definition of the 
derivative of a function of a single real variable is applicable 

* For a proof of this theorem for a closed interval, see Phillips, 
A Course of Analysis (Cambridge, 1939), p. 73. This will be referred 
to subsequently as P.A. 


to functions of a complex variable. The natural definition 
is as follows : Let f(z) be a one-valued function, defined in 
a domain D of the Argand diagram, then f(z) ia differentiate 
at a point Z Q of D if 

tends to a unique limit as z-^- z , provided that z is also a 
point of D. 

If the above limit exists it is called the derivative 
of /(z) at z = z and is denoted by /'(z ). Restating the 
definition in a more elementary form, it asserts that, given 
>0, we can find a number S such that 

(*)-/(*o) , 




for all z, z in D satisfying 0< |z z |<8. That continuity 
does not imply differentiability is seen from the following 
simple example : 

Let/(z) = | z )'. This continuous function ia differentiable 
at the origin, but nowhere else. For if z ^ we havo 

= +z (cos 2< i sin 2<f>) 

where <f> = arg (z 2 ). It is clear that this expression does 
not tend to a unique limit as z-> z . 

If Zg the incrementary ratio is z t which tends to zero 
as z-> 0. 

7. Regular Functions 

A function of z which is one-valued and differentiable 
at every point of a domain D is said to be regular * in the 
domain D. A function may be differentiable in a domain 
save possibly for a finite number of points. These points 
are called singularities of /(z). We next discuss the 
necessary and sufficient conditions for a function to be 

* The terms analytic and holomorphic are sometimes used as 
ynonymoua with the term regular as defined above. 


(1) The necessary conditions for f(z) to be regular. 

If f(z) =s u(x, y)+iv(x, y) is differentiable at a given 
point 2, the ratio {/(z+ Az) f(z)}jAz must tend to a definite 
limit as Jz->0 in any manner. Now Az = Ax+iAy. 
Take A z to be wholly real, so that Ay = 0, then 

u(x+Ax, y)u(x y y) . v(x+Ax, y)v(x, y) 
Ax + * Ax 

must tend to a definite limit as Ax-* 0. It follows that 
the partial derivatives u x , v 9 must exist at the point 
(x 9 y) and the limit is u 9 -\-iv x . Similarly, if we take 
Az to be wholly imaginary, so that Ax = 0, we find that 
t/,, v y must exist at (x, y) and the limit in this case is 
v y iuy. Since the two limits obtained must be identical, 
on equating real and imaginary parts, we get 

UX = Vy , Uy = V 9 . . . (1) 

These two relations are called the Cauchy-Riemann 
differential equations. 

We have thus proved that for the function f(z) to be 
differentiable at the point z it is necessary that the four partial 
derivatives u x , v x , u y , v y should exist and satisfy the Cauchy- 
Riemann differential equations. 

We thus see that the results of assuming differentiability 
are more far-reaching than those of assuming continuity. 
Not only must the functions u and v possess partial 
derivatives of the first order, but these must be connected 
by the differential equations (1). 

That the above conditions are necessary, but not 
sufficient, may be seen by considering Examples 6 and 7 
at the end of this chapter. 

(2) Sufficient conditions for f(z) to be regular. 
Theorem. The continuous one-valued function f(z) is 

regular in a domain D if the four partial derivatives u x , v xt 
u y , v v exist, are continuous and satisfy the Cauchy-Riemann 
equations at each point of D. 



Au = u(x+Ax y y+Ay)u(x, y), 

= u(x+Ax, y+Ay) u(x+Ax, y) +u(x+ Ax, y) u(x, y), 
= Ay . u v (x+Ax y y+OAy)+Ax . u x (x+6'Ax, y) ; 

where 0<0<1, 0<0'<1, by the mean- value theorem.* 
Since u x , u v are both continuous, we may write 
Au = Ax{u x (x, y)+<i}+Ay{u v (x, y)+c'} 9 

where and e' both tend to zero as | Az |-> 0. 

Av = Ax{v x (x, y)-i- r rj}+Ay{v 1/ (x, y)+^'}, 
where rj and rf both tend to zero as | Az |-> 0. 

Hence Aw = Au+iAv 

= AX(UB +iv x ) +Ay(u v +iv v ) +a)Ax+a)'Ay 9 

where a* and aj f tend to zero as | Az \-> 0. 

On using the Cauchy-Riemann equations we get 

Aw = (Ax+iAy)(u x +iv x )-\-a>Ax+a>'Ay 

and, on dividing by Az and taking the limit as \Az |-> 0, 

atAx+oj'Ay . . . 
since ^ I co I ~p I co I 

We notice that the above sufficient conditions for the 
regularity of f(z) require the continuity of the four first 
partial derivatives of u and v. 

If w = tt-f-iv, where u and v are functions of x and y, 

..!,., -!,-* 

* See P.A., p. 101. 


u and t; may be regarded formally as functions of two 
independent variables z and z. If u and v have continuous 
first-order partial derivatives with respect to x and y, the 
condition that w shall be independent of z is that dw/8S = 0. 
This leads to the result 

du dx 
that is 

du dy /<9v dx dv dy\ __ 
8y'8i\8x'di~dy8s)~~ ' 

I du 1 du t dv 1 dv __ ^ 

and, on writing f for 1/f and equating real and imaginary 
parts, we get 

du dv dv du 

which are the Cauchy-Riemann equations. 

Hence, in any analytical formula which represents a 
regular function of z, x and y can occur only in the combina- 
tion x+iy. For example, it is clear at a glance that 

sin (x+3iy) = sin (2zz) 

cannot be a regular function. 

If u+iv =f(x-\-iy) where f(z) is a regular function, 
then the real functions u and v of the two real variables 
x and y are called conjugate functions. 

Since the partial derivatives of u and v are connected 
by the relations 

du _ 8v 8v _ _te 

a5~V8i~~V () 

if the derivatives concerned are assumed to exist and 
satisfy the relation (f> Xv = </> vX , it follows by partial 
differentiation that 

_ _ 

"" "" an 


Hence both u and v satisfy Laplace's equation in two 

This equation occurs frequently in mathematical physics. 
It is satisfied by the potential at a point not occupied by 
matter in a two-dimensional gravitational field. It is 
also satisfied by the velocity potential and stream function 
of two-dimensional irrotational flow of an incompressible 
non-viscous fluid. 

By separating any regular function of z into its real 
and imaginary parts, we obtain immediately two solutions 
of Laplace's equation. It follows that the theory of 
functions of a complex variable has important applications 
to the solution of two-dimensional problems in mathe- 
matical physics. It also follows from equations (1) that 

The geometrical interpretation of (2) is that the families 
of curves in the (x, y)-plane, corresponding to constant 
values of u and v, intersect at right angles at all their 
points of intersection. For if u(x, y) = c l9 then du = 0, 
and so 

8 dx + ^dy = 0. . . . (3) 
ox oy 

Similarly, if v(x, y) = c 2 , we have 

The condition that these families of curves intersect at 
right angles is 



where the suffixes 1 and 2 refer to the u and v families 
respectively. On using (3) and (4), it is easy to see that 
(5) reduces to (2). 

It is possible to construct a function /(z) which has a 
given real function of x and y for its real or imaginary 
part, if either of the given functions u(x, y) or v(x, y) is 
a simple combination of elementary functions satisfying 
Laplace's equation. A very elegant method of doing this 
is due to Milne-Thomson.* 

Since x = g (2+2), y = - (z-z), 

zz . (z+z zz 

We can look upon this as a formal identity in two inde- 
pendent variables z, z. On putting z = z we get 

Now /'(z) = UX+WB = UxiUy by the Cauchy-Riemann 
equations. Hence, if we write <f>i(x t y) and <f>%(x 9 y) for 
u 9 and u y respectively, we have 

/'(z) ^^(z, y)-^ 2 (* y)=<f>i(*> 0)-i^ 2 (z, 0). 
On integrating, we have 

i(*> 0)-^ 2 (z, 0)}dz+C, 

where C is an arbitrary constant. 

Similarly, if v(x, y) is given, we can prove that 

I 2 (z, 0)}dz+C, 

where ift^x, y) = v y and 2 (#, y) = v 9 . 

Math. Gazette, xxi. (1937), p. 228. See also Misc. Ex. !, p. 138. 


As an example, suppose that u(x,y) = e*(x ooa y y sin y), 

Here fa = = e*(x cos y y sin y-fcos y), 


^ t = ~- = e*(x sin t/ sin yy cos y). 

Hence f'(z) = ^(z, 0)-t> 2 (z, 0) = e (2+!), 

and so f(z) = J e (s+l)efa+(7 = ze +O. 

8. Power Series. The Elementary Functions 

00 00 

Consider the series S a n z n or 2 a n (z z ) w , where the 

n=0 n 

coefficients a n and z, z may be complex. Since the latter 
series may be obtained from the former by a simple change 
of origin, the former may be regarded as a typical power 
series. It is assumed that the reader is already familiar 
with the theory of real power series.* 

So far as absolute convergence is concerned, everything 
that has been proved for absolutely convergent series of 
real terms extends at once to complex series, for the series 
of moduli 

Kl +lilN + KIN 2 +- 

is a series of positive terms. The most useful convergence 
test for power series is Cauchy's root test, which states 
that a series of positive terms Zu n is convergent or 
divergent according as liin (u n ) l l n is less than or greater 
than uriity.f If we write lim | a n \ l l n = 1/jR, then we 
easily see that the power series Za n z n is absolutely con- 
vergent if | z \<R, divergent if | z |>J2, and if | z | = R 
we can give no general verdict and the behaviour of the 
series may be of the most diverse nature. The number B 
is called the radius of convergence, and the circle, 
centre the origin, and radius R, is called the circle ol 

* See P.A., Ch. XIII. 
t See P.A., p. 124. 



convergence of the power series. Clearly there are three 
cases to consider (i) R = 0, (ii) R finite, (iii) R infinite. 
The first case is trivial, since the series is then convergent 
only when 2 = 0. In the third case the series converges 
for all values of z. In the second case the radius of the 
circle of convergence is finite and the power series is 
absolutely convergent at all points within this circle, 
and divergent at all points outside it. 
We now prove an important theorem. 


If f(z) = S a n z n , then the sum-function f(z) is a regular 

function at every point within the circle of convergence of 

the power series. 

Suppose that Sa n z" is convergent for | z \<R. Then, 
if 0</><J?, a^ n is bounded, say |anp n |</f. Let 

$(z) = 2 na n z*~ l . 


We write, for convenience, | z \ = r, | h \ = 77 : then, if 
r<p and 



-nan-* . 



-. = (r+^-r* _ ,-! 





KZ 1 -( (r ^ )n - rn -nr->} 

n-op n ( -n } 

l -(-r P-} /L_l 

r) \p-r-r) p-rj (p-r) 2 J 


which tends to zero as TJ-> 0. Hence f(z) has the derivative 

(f)(z). This proves that /(z), which is plainly one-valued, 

is also differentiable : hence f(z) is regular within | z \ = B. 

Since n l l n -*\ as n->oo, lim|n^ tt | 1 / n = limlaj 1 /" = 1/J2, 

and so the series <f>(z) = Z na n z n ~ l has the same radius of 


convergence as the original series. Thus, if <f>(z) = /'(z) 
is regular in | z \<R, we can show similarly that its 
derivative is Sn(n l)a n z n ~ 2 , and so on. In other words, 
we thus prove that a power series can be differentiated 
term by term as often as we please at any point within its 
circle of convergence. 

The above theorem, which is the analogue of a well-known 
theorem in real variable theory, can be superseded by a 
more general theorem which is one of the characteristic 
achievements of complex variable theory. This theorem is 
as follows : 

Let f(z) _ u l (z)+ui(z) + . . . +u n (z)+ . . . ; 

if each term u n (z) is regular within a region D and if the series 
is uniformly convergent throughout every region D' interior to 
D, then f(z) is regular within D and all its derivatives may be 
calculated by term-by -term differentiation. 

For a proof of this theorem, the reader is referred to 
larger treatises on complex variable theory. The simpler 
theorem proved above will suffice for the purposes of this 

In a later chapter ( 34) we prove Taylor's theorem 
that a function /(z) can be expanded in a power series 


2a n (za) n about any point a, provided that/(z) is regular 


in | za |<p. By combining Taylor's theorem with the 
theorem proved above, we see that the necessary and 
sufficient condition that a function /(z) may be expanded 
in a power series is that it should be regular in a region. 
The Weierstrassian development of complex variable 


theory begins by defining an " analytic function " of z aa 
a function expansible in a power series. (See 39.) 

We now consider briefly the definitions of the so-called 
elementary functions of a complex variable. 

I. Rational functions. 

A polynomial in z, a Q -\-a l z+...+a m z m 9 may be regarded 
as a power series which converges for all values of z. 
Since such functions are regular in the whole plane, rational 
functions of the type 

are regular at all points of the plane at which the 
denominator does not vanish. If we choose a point z , 
at which the denominator does not vanish, and replace 
* by Z +(z--z ), the function /(z) becomes 

A +A l (z-z )+...+A rn (z-z )< 

in which B Q ^ 0. It readily follows that /(z) may be 


expanded in a power series of the form Hc n (zz ) n . 


II. The exponential function. 

For the exponential function of a real variable, one 
method of development is to define exp a: as the sum- 
function of the power series 

and, on using the multiplication theorem for absolutely 
convergent series, we prove that * 

exp x . exp x 9 = exp (x+x'). 
* See P.A., p. 246. 


In the same way we can define eipz as the sum- 
function of the series of complex terms 

Since the series converges for all values of z, it defines a 
function regular in the whole z-plane. Such functions are 
called integral functions. 

When x is rational, exp x is identical with the function 
e* of elementary algebra, and when x is irrational we 
define e 9 to be identical with the function exp x, the 
sum-function of the power series (1) above. In the same 
way, when z is complex, we find it convenient conventionally 
to write e* for exp z. Since the formula exp z . exp 
= exp(z + f) can be proved by multiplication of series, 
whether z be real or complex, the real number e with a 
complex exponent obeys the formal law of indices of 
elementary algebra 

e'. ef = e*+f . 

Thus we may define the power e*, without ambiguity, 
by the equation 

z 2 z 8 
e = l + z +2\ + 3! + * 

and, if a is any positive number, a* denotes the value 
unambiguously determined by the formula 

a * ^ gi log*, 
where log a is the real natural logarithm of a. 

The reader should notice how far this definition is removed 
from the elementary definition " x k is the product of k factors 
equal to a?." At first sight there is no knowing what value 
belongs to a number of the form 2', but its value is uniquely 
determined by our definition. 


For real values of y we have 

00 (iij} n & f/2* oo 

e" - 27 ( -ff - 2; (-1)* ITT-, + t(-l) --y- (2) 
n-o * ! t o ( 2k ) ! *~o (2&+1) ! 

= cos y+ i sin y ; 

since the cosine and sine of the real variable y are defined 
by the two power series on the right of (2). 

Hence e* = e** iv = eV* = e* cis y. 

We also see that, since 

| & | = | cisy | = 1 , | e | = | e|| e<* \ = e, 

since e*>0. Similarly, arg e f = Iz = y. TAe function e 9 
has the period 2ni ; in other words, if k is any positive 
or negative integer, or zero, 

for, when we increase z by 27rt, y increases by 2?r and 
this leaves the values of sin y and cos y unchanged. Every 
value which e* is able to assume is therefore taken in the 
infinite strip 7r<j/<7r, or in any strip obtainable from 
this by a parallel translation. 

It is easy to show that e 9 has no other period. If 
e f = ef, this necessarily implies that z = f +2kiri. This 
follows at once, because e f ~f = 1 and so 

e*~f cis(y 77) = 1. 

Hence x % = 0, cos (yrj) = 1, sin (yrj) = ; and this 
leads to y7) = Zkn, so that z = 2&?rt. 

Finally, e* never vanishes, for e**e-*i = 1, and, if 
e* =s 0, this equation would give an infinite value for 
e~*>, which is impossible. 

Since z = z+iy = r cos 0+ir sin 0, any complex number 
may be written in the form z = re^, where | z \ = r, 
arg z = #, since we have now assigned a meaning to e'0. 


By term-by-term differentiation of the power series 
defining e* t we readily see that 

T *>* = * 


III. The trigonometrical and hyperbolic functions. 

We define sin z and cos z, when z is complex, as the 
sum-functions of power series, just as we do for sin x 
and cos x when x is real. Thus 

sin Z = - 


and, since each of these power series has an infinite radius 
of convergence, sin z and cos z are integral functions. 

By term-by-term differentiation of these power series, 
we deduce at once that the derivatives of sin z and cos z 
are cos z and sin z respectively. 

The other trigonometrical functions are then defined by 

sin z A 1 1 1 

tan z = , cot z = , sec z = , cosec z = - . 

cosz tanz cosz sinz 

If we denote exp iz by e**, according to our agreed con- 
vention, we readily obtain the results 

cos z+i sin z = e**> cos z i sin z == er** ; 
leading to Euler's formulae 

cos z = - (e' +*-*) , sin z = - (et'-e-**). 

& +->i 

From these formulae, and the addition formula for e*, 

we find that . , , 

sm 2 z+cos 2 z =5 1 ; 

and the addition theorems 

sin(z ) = sin z cos cos z sin , 
cos(z J) s=s cos z cos S^sin z sin , 

also hold for complex variables. As all the elementary 
identities of trigonometry are algebraic deductions from 


these fundamental equations, all such identities also hold 
for the trigonometrical functions of a complex variable. 

The hyperbolic functions of a complex variable are 
also defined in the same way as for real variables. The 
two fundamental ones, from which the others may be 
derived, are 

sinh z = |(e f e~*), cosh z = \(e* +e~ f ). 

These two functions are clearly regular in any bounded 

The important relations 

sin iz = i sinh z, cos iz = cosh , 
sinh iz = i sin z, cosh iz = cos z, 

are easily proved and are of great usefulness for deducing 
properties of the hyperbolic functions from the correspond- 
ing properties of the trigonometrical functions. 
If we write z = x+iy, 

sin z = sin x cosh y +i cos x sinh y, 

and we see that sin z can only vanish if 

sin x cosh y = 0, cos x sinh y = 0. 

Now coshy>l, and so the first equation implies that 
sin x is zero. Hence x = mr, (n = 0, 1, ^2, ...). The 
second then becomes sinh y = 0, and this has only one 
root y = 0. Hence sin z vanishes if, and only if, 
z = ttTr, (n = 0, 1, 2, ...). Similarly, we can show that 
cos z vanishes if, and only if, z = (n-\-%)ir. 

IV. The logarithmic function. 

When x is real and positive, the equation e" = x has 
one real solution u = log x. If z is complex, however, but 
not zero, the corresponding equation exp w = z has an 
infinite number of solutions, each of which is called a 
logarithm of z. If w = u+iv we have 

e"(cos t>+*' sin t;) = z. 


Hence we see that v is one of the values of arg z and e u = \z\. 
Hence u = log | z |. Every solution of exp w = z is thus 
of the form 

w = log |z|+iargz. 

Since arg z has an infinite number of values, there is an 
infinite number of logarithms of the complex number z, 
each pair differing by 2iri. We write 

Log z = log \z | + i arg z, 

so that Log z is an infinitely many- valued function of z. 

The principal value of Log z, which is obtained by 
giving argz its principal value, will be denoted by logz, 
since it is identical with the ordinary logarithm when z 
is real and positive. We refer again to the logarithmic 
function in the next section, where many-valued functions 
are discussed in more detail. 

V. The general power f . 

So far we have only defined a 1 when a>0. If z and 
denote any complex numbers we define the principal 
value of the power *, with ^ as the only condition, 
to be the number uniquely determined by the equation 

where log is the principal value of Log . By choosing 
other values of Log we obtain other values of the power 
which may be called its subsidiary values. All these are 
contained in the formula 

Hence f has an infinite number of values, in general, 
but one, and only one, principal value. 

Example, t* denotes the infinity of real numbers 
exp{i(logi+2kiri)} = exp {i(%7Ti+2kiri)} 
= exp ( ^TrZk-rr). 
exp (Jw) is the principal value of the power i 4 . 

If 0, Rz>0, we define f to be zero. 


9. Many-valued Functions 

In the definition of a regular function given in 7, we 
note that a regular function must be one-valued (or 
uniform). Quite a number of elementary functions, such 
as 2 (a not an integer) log z, arc sin z are many- valued. 
To illustrate the idea of many-valuedness, let us consider 
the simple case of the relation w 2 = z. On putting 
z = re*Q, w = Re i( t> 9 we get 

For given r and 9(<2rr), two obvious solutions are 
Wl = | \/r | e** 8 and w 2 = | y> I e^ e +^ = -| y> I* 4 **, 

and these are the only continuous solutions for fixed 0, 
since |yV | and | \/r \ are the only continuous solutions 
of the real equation x 2 = r, r>0. 

In particular, for a positive real z, that is when = 0, 
w = | \/r | and u> 2 = | \/ r l> an d both w l and tu 2 are 
one-valued functions of z defined for all values of z. 

If we follow the change in w l as varies from to 27r, 
in other words, as the variable z describes a circle of radius r 
about the origin, w l varies continuously and we see that 
the final value of w l is | \Jr \ e^ ' 27rt = | ^/r \ = w 2 . 

Hence the function w l is apparently discontinuous 
along the positive real axis, since the values just above 
and just below the real axis differ in sign and are not 
zero (except at the origin itself). If, however, z describes 
the circle round the origin a second time, the values of 
w l continue those of w 2 and at the end of the second 
circuit we have w 2 = w l along the positive real axis. 

We thus see that the equation w 2 = z has no continuous 
one-valued solution defined for the whole complex plane, 
but w 2 = z defines a two-valued function of z. The two 
functions w l = | \/r \ e W and w? 2 = | \/r \ e^ are 
called the two branches of the two-valued function 
w 2 = z. Each of these branches is a one- valued function 


in the z-plane if we make a narrow slit, extending from 
the origin to infinity along the positive real axis, and 
distinguish between the values of the function at points 
on the upper and lower edges of the cut. If OA = x , in 
fig. 2, the value of w l at A(6 = 0) is | \/# | and the value 

B * 

FIG. 2. 

of w l at B(6 = 2n) is | \/a? I- Since the cut effectively 
prevents the making of a complete circuit about the origin, 
if we start with a value of z belonging to the branch w v 
we can never change over to the branch w%. Thus w l (and 
similarly u> 2 ) is one-valued on the cut-plane. 

There is an ingenious method of representing the two- 
valued function w 2 = z as a one-valued function, by 
constructing what is known as a Riemann surface. 
This is equivalent to replacing the ordinary z-plane by 
two planes P l and P 2 : we may think of P l as superposed 
on P 2 . If we make a cut, as described above, in the 
two planes, we make the convention that the lower edge 
of the cut in P l shall be connected to the upper edge of 
the cut in P 2 and the lower edge of the cut in P 2 to the 
upper edge of the cut in P v Suppose that we start with 
a value z of z, t#J being the corresponding value of w l 
on the plane P 1 , and let the point z describe a path, 
starting from z , in the counter-clockwise sense. When 
the moving point reaches the lower edge of the cut in Pj 
it crosses to the upper edge of the cut in P 2 , then describes 
another counter-clockwise circuit in P 2 until it reaches 
the lower edge of the cut in this plane. It then crosses 
again to the upper edge of the cut in P 1 and returns to its 
starting point with the same value W with which it started. 
This corresponds precisely to the way in which we obtain 
the two different values of \/z, and so \/z is a one-valued 
function of position on the Riemann surface. 


There is no unique way of dividing up the function 
into branches, and we might have cut the plane along 
any line extending from the origin to infinity, but the 
point z = is distinguished, for the function w = \/z 9 
from all other points, as we shall now see. 

We observe that, if z describes a circle about any point 
a and the origin lies outside this circle, then arg z is not 
increased by 2n but returns to its initial value. Hence 
the values of w l and u> 2 are exchanged only when z turns 
about the origin. For this reason the point z = is called 
a branch-point of the function w = \/z and, as we have 
already stated, w^z) and w 2 (z) are called its two branches. 

Since turning about z = oo means, by definition, 
describing a large circle about the origin, the point z = oo 
is also a (conventional) branch-point for w = <\/z. 

The relation w n = z defines an n- valued function of z, 

since z* has n, and only n, different values 



, (s = 0, 1, 2, ..., ft 1). 

The point z = is a branch-point, and the Riemann 
surface appropriate to this function consists of n sheets 
P 19 P 2 , ..., P n - Plainly z = 1 is a branch-point for 
w = \/(z 1) and the cut is made from z = 1 to z = oo. 

For w = Log z, since w = log r+i(0-f 2for), every 
positive and negative integer k gives a branch, so Log z 
is an infinitely many- valued function of z. The Riemann 
surface consists of an infinity of superposed planes, each 
cut along the positive real axis, and each edge of each 
cut is joined to the opposite edge of the one below. The 
points z = and z = oo are branch points. 

For w = <\/{(za)(zb)} we make a cut on each plane 
along the straight line joining the points z = a and z = 6, 
and join the planes P v and P f cross- wise along the cut. 
In this case infinity is not a branch-point. 

If u>= \/{(*-~ a i)(*~ a i)---( z "~ a *)}; when k is even, 


we make cuts joining pairs of points a r , a f ; and when 
k is odd, one of these points must be joined to oo, as in 
the case k = 1. The edges of the various cuts on the 
two planes are joined cross- wise as in the case w = ^/z. 

Considerable ingenuity is required in constructing 
Riemann surfaces for more complicated functions, but it 
is beyond our scope to pursue this question further. 

Note on notation. In what follows we shall frequently 
use w, z and to denote complex numbers, and, whenever 
they are used, it will be understood, without further 
explanation, that 

w = u+iv, z SB x+iy, ss +1*77. 

Other symbols, such as t, r, s, may occasionally be used 
to denote complex numbers, but we do not specify any 
special symbols to denote their real and imaginary parts. 


1. Prove that | z l -z 9 1 2 + | z^z 9 | = 2\ z l \*+2 \ z 2 | ; and 
deduce that 

all the numbers concerned being complex. 

2. Prove that the area of the triangle whose vertices are 
the points z l9 z s , z 3 on the Argand diagram is 

{(* 2 -* 8 )|Zl |*/*l}. 

Show also that the triangle is equilateral if 

3. Determine the regions of the Argand diagram defined by 
|s-3|<l; \z-a\ + \z-b\^k(k>0); \ z*+az+b \<r*. 

In the last case, show that, if z l9 z a are the roots of z*+az+b = 0, 
we obtain two regions if r< J | z l z t |. 

4. A point P(a+ib) lies on the line AB, where A is z = p 
and B is z = 2ip. If Q is p a /(a +ib), find the polar coordinates 


of P and Q referred to the origin as pole and the real axis 
as initial line. Indicate the positions of P and Q in an 
Argand diagram. If P move along the line AB and G is the 
point z = p, prove that the triangles OAP, OQC are similar, 
and that the locus of Q is a circle. 

5. If /<) - ' <**0),/(0) = 0, 

prove that {/(z) /(0)}/z-> as z-$~ along any radius vector, 
but not as z-> in any manner. 

6. Prove that the function u+iv = /(z), where 

is continuous and that the Cauchy-Riemann equations are 
satisfied at the origin, yet/'(0) does not exist. 

7. Show that the function /(z) = \f\ xy \ is not regular 
at the origin, although the Cauchy-Riemann equations are 
satisfied at that point. 

8. If /(z) is a regular function of z, prove that 

9. If w ^/(z) is a regular function of z such that/'(z) ^ 0, 
prove that 

If \f'(z) \ is the product of a function of x and a function oft/, 
show that 


where a is a real and )3 and y are complex constants. 
10. Prove that the functions 

(i) u t 
(ii) u = sin a; cosh y +2 cos a; sinh y -\-x* t/ f 

both satisfy Laplace's equation, and determine the corre* 
spending regular function u+iv in each caae, 


11. If w = arc sin z, show that w = nrr ii 

according as the integer n is even or odd, a cross-cut being 
made along the real axis from 1 to oo, and from oo to 1 
to ensure the one-valuedness of the logarithm. 

12. If w =* V{(1-*)(1 +**)>, ^ the point (2, 0) and P 
a point in the first quadrant, prove that, if the value of w 
when z = is 1, and z describes the curve OP A, the value 
of w at A is i\/5. 

13. If w = V(2 2z+z f )> and z describes a circle of 
centre z = 1 +i and radius V2 in the positive sense, determine 
the value of w (i) when z returns to O, (ii) when z crosses the 
axis of y, given that z starts from O with the value + \/2 of w. 

14. Prove that log(l +z) is regular in the z-plane, cut along 
the real axis from oo to 1, and that this function can be 
expanded in a power series 

z* z* z* 

'-2 + 3-4 + 
convergent when | z \ < 1. 

15. Prove that the function 

-"< n+1) 


n~l "I 

is regular when | z \ < I and that its derivative is o/(z)/(l -fa). 
Hence deduce that/(z) = (l+z) a . 

16. (i) Prove that the exponential function e 9 is a one- 
valued function of z. 

(ii) Show that the values of z = a*", when plotted on the 
Argand diagram for z, are the vertices of an equiangular 
polygon inscribed in an equiangular spiral whose angle is 
independent of a. 

17. If 0<a <a 1 <...<a ll , prove that all the roots of the 

lie outside the circle | z | = 1. 

18. Show that, if 6 is real and sin sin <f> = 1, then 

where n is an integer, even or odd, according as sin 0>0 
or sin 0<0. [If ^ = o+t'0, we have sin a cosh ft = cosec 0, 
cos a sinh = 0. Solve for a and ft.] 



10. Isogonal and Gonformal Transformations 

The equations u = u(x, y), v = v(x, y) may be regarded 
as setting up a correspondence between a domain D of the 
(x, t/)-plane and a domain D f of the (u, v)-plane. If the 
functions u and v are continuous, and possess continuous 
partial derivatives of the first order at each point of D t 
then any curve in D, which has a continuously turning 
tangent, corresponds to a curve in D' possessing the same 
property, but the correspondence between the two domains 
is not necessarily one-one. 

For example, if u = x* 9 v = t/*, the circular domain 
f +2/ f <l corresponds to the triangle formed by the lines 
u = 0, v = 0, w+v = 1, but there are four points of the 
circle corresponding to each point of the triangle. 

If two curves in the domain D intersect at the point P, 
(X Q , y ) at an angle 0, then, if the two corresponding curves 
in D' intersect at the point (U Q , v ) corresponding to P 
at the same angle 0, the transformation is said to be isogonal. 
// the sense of the rotation as well as the magnitude of the 
angle is preserved, the transformation is said to be conf ormal. 

Some writers do not distinguish between isogonal and 
conf ormal, but define conformality as the preservation of the 
magnitude of the angles without considering the sense. 

If two domains correspond to each other by a given 
transformation u = u(x, y), v = v(x 9 y), then any figure in 
D may be said to be mapped on the corresponding figure 


in D' by means of the given transformation. We have 
already defined isogonal and conformal mapping, but it 
should be observed that, if one domain is mapped isogonally 
or conformally upon another, the correspondence between 
the domains is not necessarily one-one. If to each point 
of D there corresponds one, and only one, point of D', 
and conversely, the mapping of D on Z)', or of D 9 on Z), 
is said to be one-one or biuniform. 

Suppose that w = f(z) is regular in a domain D of the 
z-plane, z is an interior point of Z), and C l and C 2 are 
two continuous curves passing through the point z , and 
let the tangents at this point make angles a l9 a a with the 
real axis. Our object is to discover what is the map of 
this figure on the uj-plane. For a reason which will 
appear in a moment, we suppose thatf(z Q ) ^ 0. 

Let Zj and z a be points on the curves C l and (7 a near to 
z and at the same distance r from z q , so that 

z = re'i , z a z c = 

then, as r-> 0, 1 -> a l and 2 -> a a . 

The point z corresponds to a point t0 in the u?-plane 
and z l and z 2 correspond to points w l and u? a which describe 
curves /\ and F 2 . Let 

then, by the definition of a regular function, 

and, since the right-hand side is not zero, we may write 
it JRe<A. We have 

and BO lim (^j QI) == A or 

lim <f> l = a x -f A. 


Thus we see that the curve F l has a definite tangent 
at WQ making an angle 04 +A with the real axis. 

Similarly, F% has a definite tangent at u> making an 
angle a 2 +A with the real axis. 

It follows that the curves F l and F 2 cut at the same angle 
as the curves C l and C 2 . Further, the angle between the 
curves F l9 F^ has the same sense as the angle between O f 1 , (7 2 . 
Thus the regular function w=f(z), for which /'(z ) ^ 0> 
determines a conformal transformation. Any small figure 
in one plane corresponds to an approximately similar 
figure in the other plane. To obtain one figure from the 
other we have to rotate it through the angle A = arg (f'(z Q )} 
and subject it to the magnification 

It is clear that the magnification is the same in all directions 
through the same point, but it varies from one point to 

If is a regular function of w and w is a regular function 
of z, then is a regular function of z, and so, if a region 
of the z-plane is represented conformally on a region of 
the w-plane and this in its turn on a region of the -plane, 
the transformation from the z-plane to the {-plane will 
be conformal. 

There exist transformations in which the magnitude 
of the angles is conserved but their sign is changed. For 
example, consider the transformation 

w = x iy ; 

this replaces every point by its reflection in the real axis 
and so, while angles are conserved, their signs are changed. 
This is true generally for every transformation of the 

. (I) 


where f(z) is regular ; for it is a combination of two 

(i){~, (ii) * 

In (i) angles are conserved but their signs are changed, 
and in (ii) angles and signs are conserved. Hence in the 
given transformation, angles are conserved and their signs 
changed. Thus (I) gives a transformation which is isogonal 
but not conformal. 

We have seen that every regular function t0=/(z), 
defined in a domain in which /'(z) is not zero, maps the 
domain in the z-plane conformally on the corresponding 
domain in the u;-plane. Let us now consider the problem 
from the converse point of view. Given a pair of differenti- 
able relations of the type 

u = u(x 9 y), v = v(x, y) . . (2) 

defining a transformation from (x y y) -space to (u, v)-space 
does there correspond a regular function w = /(z) ? 

Let da and ds be elements of length in the (u, v) -plane 
and (x, y) -plane respectively. Then da 1 = du z -{-dv 2 9 
ds 2 = dx^+dy 2 and so, since 

du du 8v dv 

du = dx + ~ dy , dv = dx + dy ; 
8x dy 8x 8y 

da 2 = Edx*+2Fdxdy+Gdy\ 

l-Y F=- 8 - + - 
\dx) ' 8x By "*" 8x 8y ' 

E = 4- 


Then the ratio do : ds is independent of direction if 

E F O 


On writing A* for E (or Q), where h depends only on 
x and y and is not zero, the conditions for an isogonal 
transformation are 

'- 1 * (*)'-* 

du du dv 8v _ 
'dx dy ~8x 8y ~~ 

The first two equations are satisfied by writing u 9 = h cos a, 
v m = h sin a, u y = A cos /?, v, = A sin , and the third is 
plainly satisfied if 

Hence the correspondence is isogonal if either 

(a) u 9 = v v , v x = u v or (b) u m = v v , v x = u v . 

Equations (a) are the Cauchy-Riemann equations and 
express that u+iv = f(x-{-iy) where f(z) is a regular 
function of z. Equations (b) reduce to (a) by writing 
v for v, that is, by taking the image figure found by 
reflection in the real axis of the u;-plane. Hence (b) 
corresponds to an isogonal, but not conformal trans- 
formation, and so it follows that the only conformal 
transformations of a domain in the z-plane into a domain 
of the u?-plane are of the form w = /(z) where f(z) is a 
regular function of z. 

The casef'(z) = 0. 

We laid down above the condition that /'(2 ) = 0. 
Suppose now that/'(z) has a zero * of order n at the point z 0f 
then, in the neighbourhood of this point, 

where a = 0. Hence 

or Pl f "* = | a | r n + l 

See 36, 


where A = arg a. Hence 

Similarly, lim< 2 = A+(n+l)a 2 . 

Thus the curves F l9 F 2 still have definite tangents at 
w , but the angle between the tangents is 

Also the linear magnification, lim (pjr) 9 is zero. Hence 
the conformal property does not hold at such a point. 

For example, consider w = z*. In this arg w = 2 arg z 
and the angle between the line joining the origin to the point 
t# and the positive real axis is double the angle between 
the line joining the origin to the corresponding point z and 
the positive real axis in the z-plane. Corresponding angles 
at the origins are not equal because, at z = 0, dwjdz = 0. 

Points at which dw/dz = or oo will be called critical 
points of the transformation defined by w =/(z)- These 
points play an important part in the transformations. 

11. Harmonic Functions 

Solutions of Laplace's equation, V 2 F = 0, are called 
harmonic functions ; and, in applications to mathematical 
physics, an important problem to be solved is that of 
finding a function which is harmonic in a given domain 
and takes given values on the boundary. This is known as 
Dirichlet's problem. In the three-dimensional case, if we 
make a transformation from (#, y, z)-space to (, T?, )-space, 
it will, in general, alter Dirichlet's problem. If F(x, y, z) 
is harmonic, and we make the transformation 

* = <i( >?> y = < a ( *n> 0. ^ = <&,(, 7), )> 

the function F 1 (^ : , 77, f ), into which F(z, y, z) is transformed 
is not, in general, harmonic in (, TJ, )- 8 P ace - In two- 
dimensional problems, however, if the transformation is 


conformal, Laplace's equation in (x, y)-space corresponds to 
Laplace's equation in (u, v)-space, and the problem to be 
solved in (u, v)-space is still Dirichlet's problem. To prove 
this, consider a transformation 

u = u(x, y), v = v(x, y) . . . (1) 

where w = u+iv is a regular function of z = x+iy, say 
w = /(z). Let Z) be a domain of the (x, y) -plane throughout 
which /'(z) T 0, and let J be the domain of the (u, v)-plane 
which corresponds to D by means of the given trans- 

If x and y are the independent variables, and V any 
twice-differentiable function of x and y, we have * 

8 2 V 8 2 V 8 2 V 8V 8V 

8u . 8u _ . 8v . , 8v . 

and du = dx + dy, dv = dx + dy, . (3) 
ox cy ox oy 

. . (4) 

with a similar expression for d 2 v. 

On substituting for du, dv, d z u, d 2 v the expressions 
(3) and (4), the expression (2) for d 2 V becomes a quadratic 
expression in the differentials of the independent variables 
dx and dy and, on selecting the coefficients of dx 2 and dy 2 9 
we get 

4. 2 i 
a/ "*" 

^\ f a. 2 



dy dy av 2 \8y du dy* dv dy*' 

But we have already seen that u and v satisfy the Cauchy- 
Riemann equations 

du __ dv dv _ 8u 

lfa~ 8y' fa~~ ~"8y 9 
* See P.A., p. 233. 



and also Laplace's equation. On addition we therefore get 
8 2 F 8 2 F\ 

M 2 l 
J ' 

;rr + ^~r = 

3 2 F A 

^r + -T-T = 0. 

Now f'(z) = tt+u;, and so the last bracket is equal 
to |/'(z)| 2 , the square of the linear magnification of the 
transformation. Since this is not zero, it follows that if 


12. Superficial Magnification 

We have already seen that the linear magnification 
at any point in a conformal transformation w = f(z) is 
|/'(z)|, it being supposed that f'(z) ^0. We now prove 
that the superficial magnification is |/'(2)| 2 . If A be the 
closed domain of the uj-plane which corresponds to a 
closed domain D of the z-plane, the area A of A is given by 



by the well-known theorem for change of variables in a 
double integral.* Now 

d(u, v) _ dtt dv dv 8u _ 
d(x, y) ~~ 8x By 8x dy ~~ 
by the Cauchy-Riemann equations : but, as we have just seen, 

du . dv 2 
dx 8x 

and so m C C , , X|9J , 


dx) \8y) 

I/'WI 1 

This proves the theorem. 

* P.A., p. 302; or Gillespie, Integration, p. 
cited as Q.I.) 

40. (Hereafter 


13. The Bilinear Transformation 

We have seen that a regular function wf(z), for 
which /'(z ) 7^ 0, gives a continuous one-one representation 
of a certain neighbourhood of the point 2 of the z-plane 
on a neighbourhood of a point W Q of the u?-plane, and 
that this representation is conformal. It may be expressed 
in another way by saying that by such a transformation 
infinitely small circles of the z-plane correspond to infinitely 
small circles of the t0-plane. There are, however, non- 
trivial conformal transformations for which this is true 
of finite circles : these transformations will now be 

Let A, J3, G denote three complex constants, A, 5, 
their conjugates and 2, a complex variable and its 
conjugate ; then the equation 

(A+I)zz+Bz+Bz+C+D = Q . . (1) 
represents a real circle or a straight line, provided that 

BB>(A+A)(C+C). ... (2) 

For, if we write A = a+ia' 9 B = b+ib', C~c+ic', 
z = x+iy, (1) becomes 

a(&+y*)+bxb'y+e = 

which is the equation of a circle. It reduces to a straight 
line if a = \(A +A) = 0. If r be the radius of this circle, 


4a a 4a 2 a' 

and the circle is real provided that 

which is the same as condition (2). 

Conversely, every real circle or straight line can, by 
suitable choice of the constants, be represented by an 
equation of the form (1) satisfying the condition (2). 


The transformation 

"-jSl < 3 > 

where a, j8, y, 8 are complex constants is called a bilinear 
transformation. It is the most general type of trans- 
formation for which one and only one value of z corresponds 
to each value of w, and conversely. Since the bilinear 
transformation (3) was first studied by Mobius (1790-1868) 
we shall, following Carath6odory,* call it also a M6bius' 

The expression a8 j8y, called the determinant 
of the transformation, must not vanish. If aS /?y = 0, 
the right-hand side of (3) is either a constant or meaning- 
less. For convenience it is sometimes agreed to arrange 
that aS /?y = 1. The determinant in the general case 
can always be made to have the value unity if the 
numerator and denominator of the fraction on the right- 
hand side of (3) be divided by \/( a <$~~/ty)- 

If we write 

^ = (a/y) +(j8y-a8)(a>/y), a> = l/, { = yz+8 . (4) 

it is easily seen that (3) is equivalent to the succession of 
transformations (4). 

Now w = z+a corresponds to a translation, since the 
figure in the w?-plane is merely the same figure as in the 
z-plane with a different origin. 

Consider next w = pz, where p is real. The two figures 
in the z-plane and the w-plane are similar and similarly 
situated about their respective origins, but the scale of 
the w-figure is p times that of the z-figure. Such a trans- 
formation is a magnification. 

In the third place we consider w = ze*0. Clearly 
| w | = | z | and one value of arg w is 0+arg z, and so 
the w-figure is the z-figure turned about the origin through 

* Conformed Representation (Cambridge, 1932). 


an angle in the positive sense. Such a transformation 
is a rotation. 

Finally, consider w = 1/z. If | z \ = r and arg z = 0, 
then | w | = 1/r and arg w = 0. Hence, to pass from 
the t^-figure to the z-figure we invert the former with respect 
to the origin of the w-plane, with unit radius of inversion, 
and then construct the image figure in the real axis of the 

The sequence of transformations (4) consists of a 
combination of the ones just considered ; of these, the 
only one which affects the shape of the figures is inversion. 
Since the inverse of a circle is a circle or a straight line 
(circle with infinite radius), it follows that a Mobius' 
transformation transforms circles into circles. 

The transformation inverse to (3) is also a Mobius' 

z = f (-8)(-a)-y ^ . . (5) 

yw a 

Further, if we perform first the transformation (3), 
then a second Mobius' transformation 

the result is a third Mobius' transformation 

_ Az+B 
{ ~ Fz+A 

where AA-BF = (08 j8y)(a'8'-j8y) ^ 0. 

Since the right-hand side of (3) is a regular function 
of z, except when z = 8/y, Mobius' transformations are 

If we write the equation of a circle (1) in the form 

P=0. . . (6) 


where d and ( are real, and then substitute 

yw> a yw a 

in (6) we get an expression of the form 

Dww+Ew+Ew+F = . . . (7) 


are both real, and it is easy to verify that the coefficients 
of w and w are conjugate complex numbers. Hence (7) 
represents a circle in the u?-plane, since it is of the same 
form as (1). 

14. Geometrical Inversion 

From what we have just seen it would be natural to 
expect that there would be an intimate relation between 
Mobius' transformations and geometrical inversion. 

Let S be a circle of centre K and radius r in the z-plane. 
Then two points P and P lf collinear with K, such that 
KP . KP l = r 2 , are called inverse points with respect to 
the circle 8, and it is known from geometry that any circle 
passing through P and P l is orthogonal to S. In the case of 
a straight line 8, P and P l are inverse points with respect 
to s, if P! is the image of P in 8. If P, P l9 and K are the 
points z, Zi, and k we have 

\(z^k)(z^k)\ = r 2 , arg fo-fc) = arg (z-&), . (8) 

the second equation expressing the collinearity of the 
points K y P y P v The two equations (8) are satisfied, if, 
and only if, 

(z l -k)(z-Jc) = r 2 . . . . (9) 
If 8 is the circle 

dzz+Bz+Bz+ Q = 0, . . . (10) 


which may be written 

we see that (10) is a circle with centre B/^ and radius 


Hence equation (9) becomes 


which on simplification is 

dz l z+Bz l +Bz+(S=0. . . . (11) 

We thus get the relation between z and its inverse z l 
from the equation of S by substituting z l for z and leaving 
z unchanged. On solving (11), the transformation is 

We now prove the theorem : 

The bilinear transformation transforms two points which 
are inverse with respect to a circle into two points which are 
inverse with respect to the transformed circle. 

If z and z 1 are inverse with respect to the circle (10) 
then (11) is satisfied. Make the transformation (3) and 
let w and w l be the transformed points. We have 


z = - , z = n: r 
l a ywa 

and, on substituting these values in (11) we get an 


where the coefficients D, E, $, F are the same as those of 
(7) ; in fact we get (7) with w replaced by w v But this 
is the condition that w and w l are inverse points with 


respect to the transformed circle (7). The theorem is 
therefore proved. 

The inversion (12) can be written as a succession of 
two transformations 


t0 = Z, Zi=* -77-. 

1 Aw+B 

The first is a reflection in the real axis and the second is 
a Mobius* transformation. The first preserves the angles 
but reverses their signs ; the second is conformal. Hence 
inversion is an isogonal, but not conformal, transformation. 
Since inversion is a one-one isogonal, but not conformal, 
transformation, it is clear that the result of two, or of an 
even number of inversions, is a one-one conformal trans- 
formation, since both the magnitude and sign of the angles 
is preserved. In other words, the successive performance 
of an even number of inversions is equivalent to a Mobius* 
transformation . 

15. The Critical Points 

If the z-plane is closed by the addition of the point 
z = oo, then (3) and (5) show that every Mobius' trans- 
formation is a one-one transformation of the closed z-plane 
into itself. 

If y ^ the point w = a/y corresponds to z = oo and 
w = oo to z = S/y ; but, if y = 0, the points z = oo, 
w = oo correspond to each other. Since, from (4), 

dw aS /?y 
'dz^ (yz+S) 2 ' 

the only critical points of the transformation are z = oo 
and z = S/y. 

These two critical points cease to be exceptional if we 
extend the definition of conformal representation in the 
following manner. A function w = /(z) is said to transform 
the neighbourhood of a point z conformally into a neigh- 
bourhood of w = oo, if the function i = l//(z) transforms 



the neighbourhood of z conformally into a neighbourhood 
of t = 0. Also w = f(z) is said to transform the neighbour- 
hood of z = oo conformally into a neighbourhood of W Q 
if w = <() =/(!/) transforms the neighbourhood of 
= conformally into a neighbourhood of w . In this 
definition w may have the value oo. 

With these extensions of the definitions we may now 
say that every Mobius 9 transformation gives a one-one 
conformal representation of the whole closed z-plane on the 
whole dosed w-plane. In other words, the mapping is 
biuniform for the complete planes of w and z. 

16. Coaxal Circles 

Let a, b y z be the affixes of the three points A, B, P of 
the z-plane. Then 

2 ft * 
arg = APB, 
6 z a 

if the principal value of the argument be chosen. Let 
A and B be fixed and P a variable point. 

FIG. 3,. 

If the two circles in fig. 3 are equal, z 1? z 2 , z 8 are the 
affixes of the points P lt P 2 , P 8 and APB = 0, we see that 

Z 2 * /i 2 i ft * z*~~b 

arg - - = 7T0 , arg = , arg 

z a a 

z, a 

z 8 a 


The locus defined by the equation 

z 6 

arg = 0, . . . (1) 

z a 

when is a constant, is the arc APB. By writing 0, 
^0, 7T+0 for we obtain the arcs AP+B, AP^B 9 AP 3 B 
respectively. The system of equations obtained by varying 
from 77 to TT represents the system of circles which can 
be drawn through the points A, B. It should be observed 
that each circle must be divided into two parts, to each 
of which correspond different values of 0. 

Let T be the point at which the tangent to the circle 
APB at P meets AB. Then the triangles TPA, TBP are 
similar and 

PB BT TP *' 

Hence TAJTB = k 2 and so T is a fixed point for all positions 
of P which satisfy 

-k (2) 

z-b ~ ' * * * ( ' 

where k is a constant. Also TP 2 = Tu4 . TB and so is 
constant. Hence the locus of P is a circle whose centre 

The system of equations obtained by varying k 
represents a system of circles. The system given by 
(1) is a system of coaxal circles of the common point 
kind, and that given by (2) a system of the limiting point 
kind, with A and B as the limiting points of the system. 
If k-+ oo or if k-* then the circle becomes a point circle 
at A or B. All the circles of one system intersect all the 
circles of the other system orthogonally. 

The above important result is of frequent application 
in problems involving bilinear transformations. It may 
be used to prove that the bilinear transformation transforms 
circles into circles. 


Suppose that the circle in the t0-plane is 


1 we substitute for w in terms of z from the bilinear 

w = 


we obtain 



A' = - 

a Ay ' 


a -Ay 


and so the locus in the z-plane is also a circle. 
We may write (3) in the form 

a z+B/a 
*i\ _ 

_ _ _ 

y z+3/y ' 

and since 


represents a circle in the z-plane, the circle in the w-plane 
corresponding to it is plainly 

By taking special values of a, j8, y, S and k the boundaries 
in the u?-plane corresponding to given boundaries in the 
z-plane are easily determined. 

For example, let 0/a = t", 8/y *= * and k = 1 : since the 
locus \(zi)/(z+i)\ =* 1 is plainly the real axis in the z-plane, 
this axis corresponds to the circle | w \ = | a/y | in the t^-plane : 
this will be the unit circle if, in addition, | a | = | y |. 


17. Invariance ol the Gross-Ratio 

Let z v z 2 , z a , z 4 be any four points of the z-plane and 
let w lf u> 2 , w s , u> 4 be the points which correspond to them 
by the Mobius* transformation 


If we suppose that all the numbers z r , w r are finite, 
we have 

'~' ~ y Zf +8 y 2 .+8 ~ (y 

and hence it follows that 

(u>i w> 4 )(w 8 w 2 ) (! z 4 )(z 3 

(2 1 -Z 2 )(2 3 -2 4 ) 


The right-hand side of (2) is the cross-ratio of the four 
points z 1? z 2 , z s , z 4 , and so we have the result that the 
cross-ratio is invariant for the transformation (1). 

If equation (2) be suitably modified, it is still true if 
any one of the numbers z r or one of the numbers w f is 
infinite. For example, let z 2 = oo and w l = oo, then 

tt>3-w 2 _ z 1 z i 


Now suppose that z f , w f (r = 1, 2, 3) be two sets each 
containing three unequal complex numbers. Suppose 
first that these six numbers are all finite. Then the 

- -~ 2 ) 


when solved for w leads to a Mobius' transformation which 
transforms each point z r into the corresponding point w r . 
The determinant of the transformation has the value 



It is also clear that (4) is the only Mobius' transformation 
which does so. The result still holds, if (4) be suitably 
modified, when one of the numbers z r or w r is infinite. 

The equation (4) above may be used to find the 
particular transformations which transform one given 
circle into another given circle or straight line. A circle 
is uniquely determined by three points on its circumference 
and so we have only to give special values to each of the 
tliree sets z r , w r (r = 1, 2, 3) and substitute them in (4). 

Example. Let z t = 1, z a = i, z 3 = 1 and w l = 0, 
u>t = 1, w 3 = oo then we get, after substitution in (4), 

which transforms the circle | z \ = 1 into the real axis of 
the w-plane and the interior of the circle | z \ < 1 into the upper 
half of the u>-plane. 

The easiest way to prove this is as follows. Equation (5) 
is equivalent to 

w i 

2 ; _ . 


The boundary | z \ = 1 corresponds to | w i \ = | w+i |, 
which is the real axis of the w-plane, since it is the locus of 
points equidistant from w = i. 

Since the centre z = of the circle corresponds to the 
point w = i, in the upper half of the w-plane, the interior 
of the circle | z \ = 1 corresponds to the upper half of the 

Similarly, since w = i corresponds to z = oo, the outside 
of the circle | z \ = 1 corresponds to the lower half of the 

It may be observed that although this use of the 
invariance of the cross-ratio will always determine the 
Mobius' transformation which transforms any given circle 
into any other given circle (or straight line), it is not 
necessarily the easiest way of doing so. 



Thus, in the previous example, since z = 1 and 1 
correspond to w =* and oo, the transformation must take 
the form 



W =3 

t corresponds to w = 1, we can determine k : thus 

Since z 

from which it readily follows that k =* i, and we obtain (5) 

18. Some special Mdbius' Transformations 

I. Let us consider the problem of finding all the Mobius* 
transformations which transform the half -plane I(z)^0 into 
the unit circle \w\ ^1. 

We observe first that to points z, z symmetrical with 
respect to the real axis correspond points w, l/w inverse 
with respect to the unit circle in the w-plane. (See 
14.) In particular, the origin and the point at infinity 
in the w?-plane correspond to conjugate values of z. Let 
the required transformation be 

w = 


Clearly y^O, or the points at infinity in the two planes 
would correspond. Since w = 0, w = oo correspond to 
z = j8/a, z = S/y we may write j8/a = a, 8/y = a 

a 2 a 

w = -- 1 
y z a 

The point z = must correspond to a point on the circle 
| w | = 1, so that 







hence we may write a = ye*0, where is real, and obtain 

w = 



Since z = a gives to = 0, a must be a point of the upper 
half-plane, in other words, Ia>0. With this condition, 
(1) is the transformation required. 

II. To find all the Mobius* transformations which 
transform the unit circle \ z (^ 1 into the unit circle \w\^l. 


w = 


In this case w = and w = oo must correspond to inverse 
points z = a, z = I/a, where | a |<1. Hence j8/a = a, 
I/a, and so 

aa za 

w = - 

y 2 I/a y dz 1* 
The point z = 1 corresponds to a point on | u; | = 1, and so 

aa 1 a 



= 1. 

It follows that, if is real, aa = ye*0 and so 

za . 

w - i v 

This is the desired transformation ; for, if z = e*V, a 

If z = re*V, where r<l, then 

r 2 ~2r6 cos (- 

-r cos ~ 



hence | w \<1 ; in other words, the interiors of the circles 

The identical transformation w = z is a special case of 
the above : if the point 2 = corresponds to w = 0, then 
a = and the transformation reduces to 

w = zeiO. 

If, in addition, dw/dz = 1 when z = we get 

w = z. 

III. The reader should find it easy to verify that the 

w== p(z-a) 

maps the circle \ z \ = p on the unit circle \ w \ = 1. If 
| a |<p it maps | z \<p on | w |<1 and | z \>p on | w 
If | a | >/> it maps | z \ >p on | w |< 1 and | z \<p on | w \ 

IV. Representation of the space bounded by three circular 
arcs on a rectilinear triangle. 

Consider three circles in the z-plane intersecting at 
the point 2 = a. 

B' (w-o) 


Fio. 4. 


The angles of the curvilinear triangle BCD of fig. 4 
are such that a+jS-f y = TT. Consider the transformation 


w = K 





The point A(z = a) corresponds to w = oo and B(z = 6) 
corresponds to w = 0. Equation (1) may be written 

. k(a-b) 

wk = , 


or w 9 = A/2', where w 9 = wk, z f = za and A = k(a b). 
Since the changes of variable from w to w' and from z to z' 
are mere translations, (1) is a pure inversion and reflexion. 
Since z = a corresponds to w = oo and each of the three 
circular arcs BC, CD, DB passes through A they correspond 
to three straight lines in the u;-plane. The two arcs BC, BD 
which pass through B correspond to two straight lines 
passing through w = 0, and the arc CD to a straight line 
which does not pass through w = 0. 

It readily follows from (1) that the shaded areas of 
fig. 4 correspond. 

With the usual convention of sign, we regard a motion 
round a closed simple contour, such as a circle, in the 
clockwise sense as positive for the area outside and negative 
for the area inside the contour. 

If, by any conformal transformation, three points 
A, B, C on a closed contour in the z-plane correspond to 
the three points A', B', C' in the w- plane lying on the 
corresponding closed contour, then the interiors correspond 
if the points A', B', C 9 occur in the same (counter-clockwise) 
order as the points A, B, C. 

We can see in this way that the shaded areas in fig. 4 
correspond. It also follows that the curvilinear triangle 
formed by the arcs AC, CD, DA in the z-plane corresponds 
to the portion of the u>-plane A'C'D'A' where A 9 is the 
point at infinity. 


1. (i) Prove that, if u = a? 1 y 1 , v = y/(#*+y f ), both 
u and v satisfy Laplace's equation, but that u+iv is not a 
regular function of z. 


(ii) Show that the families of curves u = const, v = const, 
cut orthogonally if u = x*/y, v = # 2 +2?/ 2 but that the 
transformation represented by u+iv is not conformal. 

2. Prove that, if w = x+iby/a, 0<a<6, the inside of the 
circle x* +y* = a 1 corresponds to the inside of an ellipse in 
the to-plane, but that the transformation is not conformal. 

3. Prove that, for the transformation w* = (z a)(z /J), 
the critical points are z = o, z = jS, z = J(a + j3), w = 0, 
w = $i(a-p). 

Show also that the condition that z = oo is not a critical 
point of the transformation w = /(z) is that lim z 2 /'(z) must 


be finite and not zero. 

4. If, by the inversion transformation x = k*(/p* 9 

y = fcV P J , * = 2 //> 2 where r P = k *> f2 = z 2 +</ 2 +z*> 
p* = ^-(-^-[-^ ^ e twice-differentiable function V(x, y, z) 
becomes V^f, TJ, ) prove that if 

(d a /df 2 + a 2 /^ 2 + av^D FI=O, then 

(d*/dx* + B*ldy* + B*/dz 2 ) (V/r) = 0. 

5. If w = cosh z, prove that the area of the region of the 
u;-plane which corresponds to the rectangle bounded by the 
lines x = 0, x = 2, y = 0, y = JTT is (TT sinh 4 8)/16. 

6. If a is real and O<C<TT, find the area of the domain 
in the u;-plane which corresponds by the transformation 
w = e* to the rectangle a c^.x^.a+c 9 c<t/<c. Find 
the ratio of the areas of the two corresponding domains and 
prove that the ratio ->- e 20 as c -> 0. 

7. Show that, if the function w =/(z), regular in | z \< R, 
maps the circle | z \ = r<R on a rectifiable curve G in the 
u>-plane, then the length of C is given by 





Show that the length of the curve into which the semi- 
circular arc | z | = 1, Jn"^argz<i7r is transformed by 
w = 4/(l+z) 2 is 2V2+2 log(l + \/2). (See 22, equation (4)). 

8. Find the Mobius' transformations which make the 
sets of points in the z-plane (i) a, 6, c, (ii) 2, 1 -ft, 
to the points 0, 1, oo of the u^ane. In case (ii) 


sketches the domains of the u?-plane and 2 -plane which 

9. Find a M6bius* transformation which maps the circle 
1*1^1 on | w 1 | ^ 1 and makes the points z = , 1 
correspond to w = , respectively. Is the transformation 
uniquely determined by the data ? 

10. Find the transformation which maps the outside of 
the circle | z \ = 1 on the half -plane Ru;>0, so that the 
points 2 = 1, i, I correspond to w = i, 0, i respectively. 
What corresponds in the w- plane to (i) the lines arg z = const., 
| z | ^ 1, (ii) the concentric circles | z \ = r, (r>l) ? 

11. Prove that w = (l+iz)/(i+z) maps the part of the 
real axis between 2 = 1 and z = 1 on a semicircle in the 

Find all the figures that can be obtained from the originally 
selected part of the axis of x by successive applications of this 

12. Find what regions of the w-plane correspond by the 
transformation w = (z i)/(z+i) to (i) the interior of a circle 
of centre z = t, (ii) the region t/>0, x>0 9 \z+i\ < 2. 
Illustrate by diagrams. Show that the magnification is 
constant along any circle with z = i as centre. 

13. Let GI, | zz l | =5 T! and C 8 , | z z % \ = r 8 be two 
non -concentric circles in the 2-plane, C l lying entirely within 
C f . Show that, if z = a, 2 = 6 are the limiting points of 
the system of coaxal circles determined by C l and <7 a , then 
w = k(z 6)/(2 a) transforms C l and (7, into concentric 
circles in the w -plane with centres at w = 0. If the radii 
of these concentric circles are p l and p 2 , show that, although 
there is an infinite number of such representations, p t : p f 
is a constant. 

14. Prove that, if w = (oz + 0)/(y2 + 8) and a8 0y = 1, 
then the linear and superficial magnifications are | y2 + S j" 1 , 

Show that the circle | y2 + 8 | = 1 (y ^ 0) is the complete 
locus of points in the neighbourhood of which lengths and 
areas are unaltered by the transformation. Prove that 
lengths and areas within this circle are increased and lengths 
and areas outside this circle are decreased in magnitude by 
the transformation. (This circle is called the isometric circle.) 



19. Introduction 

The fundamental problem in the theory of conformal 
mapping is concerned with the possibility of transforming 
conforrnally a given domain D of the z-plane into any 
given domain D' of the w-plane. It is sufficient to consider 
whether it is possible to map conforrnally any given 
domain on the interior of a circle. For if = f(z) maps 
D on | C|<1 and w = F(Q maps D' on | |<1, then 
w = F{f(z)} provides a conformal transformation of D 
into D'. 

The fundamental existence theorem of Riemann states 
that any region with a suitable boundary can be conforrnally 
represented on a circle by a biuniform transformation. Rigor- 
ous proofs of this existence theorem are long and difficult, 
and it is beyond our scope to discuss the question here. 

In the applications of conformal transformation to 
practical problems, the problem to be solved is as follows : 
given two domains D and D' with specified boundaries, 
find the function w = f(z) which will transform D into D' 
so that the given boundaries correspond. Although, by 
Riemann 's existence theorem, we can infer the existence of 
the regular function /(z), the theorem does not assist 
us to find the particular function f(z) for each problem 
whose solution is desired. We have seen that when the 
two domains D and D' are bounded by circles, it is fairly 
easy to find the Mobius' transformation which maps D 
biuniformly on Z)'. Since for any arbitrary boundary 
curves there is no general method of finding the appropriate 


regular function f(z), it is important to know the types of 
domain which correspond to each other when f(z) is one 
of the elementary functions or a combination of several 
such functions. 

In this chapter we discuss some of the most useful 
transformations which can be effected by elementary 
functions. The reader, who is mainly interested in the 
application of these transformations to practical problems, 
will find the special transformations discussed here of 
great value, but he must refer to other treatises for the 
details of the practical problems to which they can be 

Many useful transformations are obtained by combining 
several simple transformations. 

For example, the transformation 

seems at first sight somewhat complicated, but on examination 
it is seen to be a combination of the successive simple trans- 

formations, Z = z 8 , == - - , t = *, w == . 

1 6 t-\-l 

It can be shown that (1) maps the circular sector | z j< 1, 
0<arg z< JTT, conformally on the unit circle * | w |< 1. 

20. The Transformations w = z n 

Let w = u+iv = pety, z = x+iy = re^, then it follows 
at once that p = r n , < = n6, so that 

u+iv = r n (cos nO+i sin n0). 

From the equations 

u = r n cos ri0, v = r n sin n0, . . (1) 
* CJompare 24 IV, equation (8), with a }. 


either or r may be eliminated, giving 

f . . (2) 

or tan n0 = - . . . (3) 

u ^ ' 

Equation (2) shows that the circles r = c of the z-plane 
and the circles p = c n correspond, and in particular, that 
points on the circle r = | z \ = 1 are transformed into 
points in the to-plane at unit distance from the origin. 
The lines = const., radiating from the origin of the 
z-plane, are transformed into similar radial lines <j> = const. 
It should be noticed, however, that the line whose slope 
is in the z-plane is transformed into the line whose slope 
is n0 in the w-plane. Since z = is a critical point of the 
transformation, the conformal property does not hold at 
this point. 

In the simple case w = z 2 , the angle between two radial 
lines in the z-plane is doubled in the w-plane. The case 
w = z 2 is typical, and we shall now consider it in greater 

We consider first the important difference between the 
transformation w = z 2 and the Mobius' transformations 
discussed in the preceding chapter. In the latter, points 
of the z-plane and of the t0-plane were in one-one 
correspondence. For w = z 2 , to each point z there 
corresponds one and only one point W Q = z^, but to a 
point w Q there correspond two values of z, z= JV^ol* 
z = \\/w \. If we wish to preserve the one-one corres- 
pondence between the two planes, we may either consider 
the w-plane as slit along the real axis from the origin to 
infinity, or else construct the Riemann surface in the 
t0-plane corresponding to the two-valued function of w 
defined by w = z 2 . The method of constructing the 
Riemann surface was described in 9. 

If we use the cut w-plane, then the upper half of the 
-plane corresponds to the whole cut u?-plane. There is 


a one-one correspondence between points of the upper 
half of the z-plane and points of the whole u?-plane, and 
a one-one correspondence between points of the lower 
half of the z-plane with points of the whole t0-plane ; but 
if we choose one of the two branches of w = z 2 , say w v 
the cut plane effectively prevents our changing over, 
without knowing it, to the other branch w 2 . The positive 
real z-axis corresponds to the upper edge and the negative 
real z-axis to the lower edge of the cut along the positive 
real axis in the u;-plane. 

If we use the two-sheeted Riemann surface in the 
w?-plane, the sheet P l corresponds to the upper half of the 
z-plane for the branch w l9 and the sheet P 2 corresponds 
to the lower half of the z-plane for the branch w 2 . Thus 
there is a one-one correspondence between the whole 
z-plane and the two-sheeted Riemann surface in the 

For w = z n , where n is a positive integer, a wedge of 
the z-plane of angle 2n/n corresponds to the whole of the 
t0-plane. If we divide up the z-plane into n such wedges, 
each of these corresponds to one of the n sheets of the 
n-sheeted Riemann surface in the w-plane. 

If, for w = z 2 , we cut the u?-plane along the negative 
real axis, then the sheet P l of the Riemann surface 
corresponds to the half-plane Rz^O, and the sheet P 2 to 
the half-plane Rz<0. 

21. Further Consideration ol w = z 2 
From the equations 

w = u-}-iv = (x-\-iy) 2 = x 2 - y z +2ixy, 
we have 

u = x 2 i/ 2 , v = 2xy. . . (I) 

By regarding u and v as curvilinear coordinates of points 
in the z-plane, the transformation w = z 2 can be examined 
from a knowledge of the curves in the z-plane which 



correspond to constant values of u and v. This method 
is frequently used in applying the theory of conformal 
transformation to practical problems. 

Equations (1) show that the curves u = const., v = const, 
in the z-plane are two orthogonal families of rectangular 

The reader will easily verify that the shaded area in 
the z-plane of fig. 5 between two hyperbolas 2xy = v lt 


FIG. 5. 


2xy = v 2 corresponds to the infinite strip of the t^-plane 
shaded in the figure. Hence w = z a maps the region 
between two hyperbolas on a parallel strip. 

If B l is at infinity, the point B\ is also at infinity, 
and the interior of the hyperbola ABC is transformed 
into the part of the upper half-plane above the line 
A' B' C'. 

We also observe that the transformation w = z* makes 
circles \ za \ = c, (a, c real) 9 in the z-plane correspond to 
limaqons in the w-plane. 

Consider the circle 

2 a = ce*"*, ... (2) 

t/* -a 2 + c 2 == 2c(c cos 9 f 


Hence, on writing w a 2 -j-c a = Be^ 9 so that the pole in 
the w-plane is at w = a 2 c 2 , the polar equation of the 
curve into which (2) is transformed is the Iiina9on 

B = 2oc+2c 2 cos 0. 

When a = c the Iima9on becomes a cardioid. This is 
the case if the circle (2) touches Oy at the origin, 

22. The Transformation w = y'* 

Prom the equations 

u*v* = x, 2uv = y, . * (1) 
we get 

. . (2) 

By means of the first of the equations (2), to the straight 
lines u = const, correspond parabolas with vertex at 
x = w a and focus at the origin of the z-plane. To the 
orthogonal system of straight lines v = const., we see, 
by the second of the equations (2), there corresponds 
another system of confocal parabolas with vertex at 
x = v 2 . 

Consider the particular parabola of the first system 
corresponding to the value u = 1, 

y 2 = 4(l-*). ... (3) 

Its transform in the w-plane is the line through the point 
w = 1 parallel to the t;-axis. The points, A, B, C in 
fig. 6 correspond, in that order, to the points A', B', C' . 
The reader can easily verify that the shaded areas 
correspond, the two parabolas drawn corresponding to 
values u = 1, and u = ^ (>1). 

If w -> oo, the region developed in the 2-plane is the 
area outside the parabola (3), which accordingly corresponds 
to that part of the t#-plane to the right of the line u = 1. 

If the parameter u tends to zero, the parabola 
y 2 = 4u 2 (u 2 x) narrows down until it becomes a slit 
along the negative real axis OX 19 which is a branch-line. 



Hence the portion of the w-plane between the line u = 1 
and the line u = corresponds to the portion of the 
z-plane between the parabola ABC, y* = 4(1 x), and 
the cut along the negative real axis OX l from the origin 
to --00. 


FIG. 6. 

c f 

Hence we see that the portion of the w-plane 
corresponds to the whole z-plane cut along the negative real 
axis from to -co. 

The simple w-plane is associated in a one-one corre- 
spondence with a two-sheeted Riemann surface covering 
the z-plane. The two sheets of the Riemann surface 
would be connected along the edges of the cuts along the 
negative real axis of the z-plane in the usual way. 

The line u = 1 plainly corresponds to the same 
parabola y a = 4(l x) as does the line u = 1. Hence 
the portion of the w-plane to the left of the line u =* 1 
corresponds to the region outside the parabola ABC 
which lies on the second sheet of the Riemann surface. 

If we combine w = y'z with a Mobius' transformation 
by writing = (2/w)-~ 1 we see that the transformation 

C---1 (4) 

fc / * v*; 

transforms the region outside the parabola (3) into the interior 
of the unit circle in the {-plane. The points z = 1, z = 4, 


z = oo correspond to the points = 1, = 0, {=1. 
The focus of the parabola (3), z = 0, lies outside the region 
of the z-plane which is under consideration, and it 
corresponds to the point = oo outside the unit circle 


The reader should observe, however, that the preceding 
transformation cannot be used to represent the inside of 
the parabola ABC on the inside of the unit circle. 

The transformation w = \/z just considered, illustrates 
an important point in the use of many-valued functions for 
solving problems in applied mathematics. The transformation 
w = ^z could be used to deal with a potential problem in which 
the field was the region outside the parabola ABC of fig. 6, but 
it could not be used for a problem in which the field was the 
space inside this parabola, since two points close to each 
other, one on each edge of the cut along the branch-line OX l9 
will transform into two points on the axis of v, one in the upper 
and the other in the lower half -plane. Since these points 
are not close together in the t-0-plane, they would correspond 
to different potentials. It is important to realise that we 
cannot solve potential problems by using transformations 
which require a branch -line to be introduced into that part 
of the plane which represents the field. 

23. The Transformation w = tan 2 (j7r\/z) 

We have just seen that w = (2/\/z) l cannot be used 
to map the region inside the parabola t/ 2 = 4(1 x) on the 
unit circle |t0|^l. We now consider a transformation 
which enables us to do this. 

The transformation can be considered as a combination 
of the three transformations 

w = 

where w = u+iv, = +^'77, t = a-ftY, z = x+iy. 
The first transformation can be written 

"*" l+cos* 



If we consider the infinite strip between the lines = 0, 
= J^TT of the -plane, we see that, by writing f = far -\-irj, 
cos = i sinhrj and | w \ = 1. Thus, as y goes from 
oo to oo along the line g = far, w describes the unit- 
circle once. By writing f = irj , cos = cosh rj and w; is 
real. Thus as 77 goes from +00 to 0, w goes from 1 to ; 
and as 77 goes from to oo, w retraces its path from 
to 1. Thus the strip = 0, = \n corresponds to 
the cut-circle as illustrated in fig. 7. It is easy to verify 




z -plane 

FIG. 7. 

that the interiors correspond. The strip in the 2-plane 
is plainly that between the lines a = and a = 1. As we 
have already seen, t = \/z transforms the strip in the -plane 
into the region inside the parabola ABC, f/ 2 = 4(l x), 
with a cut from the origin to infinity along the negative 
real axis. In fact, as or -> the parabola y 2 = 4a 2 (or 2 x) 
becomes a very narrow parabola which is the slit illustrated 
in the z-plane in fig. 7. 

The transformation w = tan 2 (7r\/z) represents the 
region inside the parabola ABO on the inside of the unit 
circle | w \ = 1 in a one-one correspondence, for the real 
axis of the w-plane between 1 and corresponds to the 
real axis of the z-plane between oo and 0. The cuts in 
the z-plane and w-plane are not needed for the direct trans- 
formation from the w>-plane to the z-plane, but they are 
needed for the subsidiary transformations used in order to 
show how the boundaries of the various regions correspond. 


Since dw/dz = 77 tan(}rr\/z) sec 2 ( j7r\/ 2 )/4 \/ z > w ^ich tends 
to a finite non-zero limit as z-> 0, the points z = and 
u> = are not critical points of the transformation, and 
so the representation is conformal as well as one-one. 

24. Combinations of w = z a with Mobius' Trans- 

I. Semicircle on half-plane or circle. 

Consider the transformation 
/ z _ j c \ 2 

' (creal) - (1) 

This is clearly a combination of 

w = 2 and = (z ic)/(z+ic). 

The second of these may be written * 

. +1 

for which it is clear that the circle | z | = c corresponds to 
the imaginary axis of the -plane |-fl| = | 1|. 

The boundary of the semicircle in the z-plane A DCS A 
plainly corresponds to A'D'C'B'A' in the -plane, C' being 
the point = -oo. The sense of description of the 
two boundaries shows that the shaded areas correspond. 

Now consider w = a : if w = pety, = re*0, we have 

The shaded domain of the -plane corresponds to 
7r<0<37r/2 and so the domain of the w-plane corresponding 
to this is 27r<<<37r, which is of course the same as 
r, or the upper half of the w-plane. 

* The use of the results of 16 is frequently simpler than the 
procedure of splitting up the transformation into its real and 
imaginary parts. 



Hence the interior of the shaded semicircle of fig. 8 
corresponds to the upper half of the w-plane. 


FIG. 8. 

It is easy to verify that the upper half of the w-plane 
corresponds to the interior of the respective semicircles 
BAEDB, AECDA, ECBDE by the transformations 

w = I 1 , w 

w - 

Also, by combining (1) with the transformation 
the transformation 


t == 


t = 


i + (*!=*]* 

. z 


2 a C 2 2CZ 

conformally represents the interior of the z-semicircle ABCDA 
on the interior of the unit circle 1 1 1 = 1. 

II. Wedge or sector on half -plane. 
By the transformation 

w = 2 1 / , 



the area bounded by the infinite wedge of angle na with its 
vertex at z = and one arm of the angle along the positive 
x-axis is transformed into the upper half of the w-plane. 
The reader will find this quite easy to verify. 

The sector cut off from this wedge by an arc of the unit- 
circle | z | = 1 is transformed by (2) into the unit semicircle 
in the upper half of the w-plane. 

This is also easy to verify. 

III. Circular crescent or semicircle on half -plane. 

We readily see, from 18, IV, that the circular crescent 
with its points at z = a and z = b and whose angle is 
TTOL can be transformed into the wedge mentioned in II 
above by 

z ~ a 

if the constant k be suitably chosen. Hence the crescent 
can be transformed into the w-half-plane by 


=} i) () 

A semicircle may be regarded as a particular case of a 
crescent in which a = J. The semicircle of radius unity 
and centre 2 = lying in the upper half-plane is trans- 
formed into the first quadrant of the -plane by 

The quarter-plane becomes a half-plane by * w = 2 and 
so the semicircle is transformed into the upper half of the 
w-plane by 

=<=()' < 6) 

* See I above. 



IV. Sector on unit circle. 

Consider the sector in the z-plane, shaded in fig. 9. 
Let us find the transformation which represents this sector 
on a unit circle. 

o j 

Fia. 9. 

By means of (2) the sector is transformed into the unit- 
semicircle in the upper half of the w-plane. By means 
of (5) we see that this unit semicircle is transformed into 
the upper half of the J-plane by 


Again, the upper half of the -plane is transformed into 
the interior of the unit circle in the -plane, | f | = 1, by 

.... (7) 

l-fz 1/a \ 2 

and on combining these, the transformation which represents 
the shaded area of fig. 9 on the unit circle in the {-plane is * 

V. By combining w = \/z with a Mobius' transforma- 
tion we find in a similar way the transformation which 
represents the z-plane, cut from to oo along the positive 
real axis on the unit circle \ | < 1 in the form 

When a = J, this transformation is the same as (1) of 19. 


VI. Transformations of the cut-plane. 
Consider the two transformations 

z a za 

W = r , W = r 

zb zb 

where a and b are real and a> 6. By means of the first 
of these, the z-plane, cut along the real axis from z = a 
to +00 and from z = b to oo, is transformed into the 
usplane cut from w = to w = oo, the cut passing through 
the point w = 1 which corresponds to z = oo. By means 
of the second, the 2-plane cut from z = a to z = 6 is 
transformed into the u;-plane cut along the positive real 
axis from to oo. The cut in this case does not pass 
through w = 1, the point corresponding to z = oo. 

25. Exponential and Logarithmic Transformations 

Most of the transformations so far considered have 
been Mobius' transformations, w = z a and combinations 
of these two types. We now observe that the relation 

t* = e . . . . (1) 

gives rise to two important special transformations. 

If we use rectangular coordinates x, y and polar co- 
ordinates />, <f> in the to-plane we get 
p = e * , <f> = y. 

The horizontal strip of the 2-plane bounded by the 
lines y = y l and y = y 2 where | y^y^ |<27r is transformed 
into a wedge-shaped region of the t^-plane, the angle of 
the wedge being a = |< 2 ~~^i I ^ I V^^Vi I- The repre- 
sentation is conformal throughout the interior of these 
regions since dw/dz is never zero. In particular, if y l = 0, 
y 2 = TT, so that | y^y\ \ = TT, the wedge becomes a half- 
plane. The semi-infinite strip oo<#<0, 0<y<7r is 
readily seen to correspond to unit semicircle in the upper 
half of the u>-plane. 

If | yiy^ |>2rr, the wedge obtained covers part of the 


w-plane multiply. We may in this case make use of the 
cut w-plane. If y l = , y a = 27r, the strip of width 2ir 
in the z-plane corresponds to the w-plane cut along the 
positive real axis. When | s^ 1/ 2 \ is an integral multiple 
of 277 the strip is transformed into a Riemann surface. 
Each strip of the z-plane of breadth 2rr corresponds to one 
sheet of the oo-sheeted Riemann surface. 

A second special transformation is obtained from (1) 
by considering an arbitrary vertical strip bounded by the 
lines x = x l , x = x 2 , (x^x^. This strip is represented 
on a Riemann surface which covers the annulus between 
the concentric circles \w\=p l ,\w\=p 2 an infinite 
number of times. If we keep # 2 constant and let x l -^~ oo, 
the strip x l <x<x 2 becomes in the limit the portion of 
the z-plane to the left of the line x = # 2 , and we obtain 
in the w-plane a Riemann surface which covers the circle 
| w |</> 2 except at the point w = 0, where it has a 
logarithmic branch-point. 

The inverse function 

w = Log z . . . (2) 

gives, on interchanging the z-plane and u;-plane, exactly 
the same transformations as (1). 

It should be remembered that although Log z is an 
infinitely many- valued function of z, e* is one-valued. 

Since z a = e a Log * the transformation w = z a may be 
regarded as a combination of the two transformations 

w = ef , = a Log z. 

26. Transformations involving Confocal Conies 

Consider the transformation 


If to = reW, we get 

2x = |(a 6)r + ^tH cosl9, 2y = j(a-6)r- a \ sin 0, 


and so the curves in the z-plane, corresponding to concentric 
circles in the w-plane having the origin for their centre, 
are confocal ellipses, the distance between the foci being 
2<v/(a 2 6 2 ). The curves in the z-plane corresponding to 
straight lines through the origin in the u;-plane are the 
confocal hyperbolas, a result to be expected, since the 
two families of curves in each plane must cut orthogonally. 
Clearly there is no loss of generality by taking a = 1, 
6 = 0, and so we may consider the transformation 

2z = w + - . . . (2) 


as typical. Clearly z becomes infinite when w = 0, and 

dw ~~" : 

the points w = 1, at which the derivative vanishes, are 
critical points of the transformation. We now have 

2x = I r -f- ~ I cos , 2y = I r I sin 

and on eliminating 0, we get the ellipse in the z-plane 

rr 2 */ 2 

= 1 , . . (3) 

corresponding to each of the two circles | w \ r , | w \ = 1/r. 
As r->l, the major semi-axis of the ellipse tends to 1, 
while the minor semi-axis tends to zero. As r-> 0, or as 
r-> oo, both semi-axes tend to infinity. From this it is 
plain that the inside and the outside of the unit circle in 
the w-plane both correspond to the whole z-plane, cut 
along the real axis from 1 to 1. The unit circle | w \ | = 1 
itself corresponds to a very narrow ellipse, which is the 
cut along the real axis enclosing the critical points 1 and 1. 


On solving equation (2) for w we get 

w = zA/(z 2 -l) 

and the inverse function is a two-valued function of z. 
If we choose the lower sign, the transformation 

w = z -V(z*-l) ... (4) 

transforms the area outside the ellipse (3) conformaHy into 
the inside of the circle \ w \ = r. The lower sign is the 
correct one to select, since the point w = inside the circle 
must correspond to the point z = oo ; the other sign of 
the square root would make the points w = oo, z = oo 
correspond. The region between two confocal ellipses in 
the z-plane is transformed into the annulus between two 
concentric circles in the w-plane. 

The function (4) gives a transformation of the z-plane, 
cut along the real axis from 1 to 1, on the interior of the 
circle \ w \ = 1. 

If we take the other sign, it is clear that the trans- 

w = 

gives a transformation of the cut z-plane on the outside of 
the unit circle \ w \ = 1. The relation (2) is remarkable 
in that it represents the cut z-plane not only on the interior 
but also on the exterior of the unit circle | w \ = 1. 

The ambiguity can be removed from this transformation 
by replacing the z-plane by a Riemann surface of two 
sheets, each cut from 1 to 1 and joined crossways along 
the cut. Then, of course, the interior of the unit circle 
| w | = 1 corresponds to one sheet and the exterior of the 
unit circle to the other sheet of this Riemann surface. 

If r>l, the transformation (2) maps the exterior of 
the circle | w \ = r, or the interior of the circle | w \ = 1/r, on 
the exterior of the ellipse (3). It should be observed, 
however, that the interior of the ellipse cannot be 
represented on the interior of the unit circle by any of the 


elementary transformations so far employed. We may 
remark, however, that the upper half of the ellipse (3) 
is represented by (2) on the upper half of an annular 
region cut along the real axis : this last area, and hence 
the semi-ellipse also, can be transformed into a rectangle 
by a method similar to that described in 25. The 
transformation which maps the interior of an ellipse on 
a unit circle involves elliptic functions. 

27. The Transformation z = c sin w 
From the relation 

z = csin;, (c real) . . (1) 

we get, on equating real and imaginary parts, 
x = c sin u cosh v, y = c cos u sinh v, 

so that, when v is constant, the point z describes the 

2 2 

i /o\ 

' * * l ' 

C 2 cosh 2 t; c 2 sinh 2 i; 

which, for different values of t>, are confocal ellipses. 
Consider a rectangle in the w-plane bounded by the lines 
u = i^j v = A. For all values of u, cos u is positive ; 
hence when v = A, y is positive and x varies from c cosh A 
to c cosh A, that is, the half of the ellipse on the positive 
side of the axis of x is covered. 

Let u = 77-, then y = and x = c cosh v. Hence 
as v varies from A through zero to A along the side of the 
rectangle, x passes from A' to the focus H (see fig. 10) 
and back from H to A'. 

When v = A then z describes the half of the ellipse 
on the negative side of the axis of x. When u = far then 
y = and x = c cosh v, so that z moves from A to the 
focus S and back from 8 to A. 



Hence the curve in the z-plane corresponding to the 
contour of the rectangle in the w-plane is the ellipse with 
two slits from the extremities of the major axis each to 
the nearer focus. It is easy to see that the two interiors 

FIG. 10. 

Since sin w = COS(JTT w), the transformation given by 
z = c cos w 

can be dealt with in a similar way. The details are left 
to the reader. 

The function inverse to (1), 

w = arc sin (z/c), 

is an infinitely many- valued function of z. If we use the 
cut plane, the cuts must be from S to infinity along the 
positive real axis and from H to infinity along the negative 
real axis. The Riemann surface of an infinite number of 
sheets in the z-plane, which would secure unique corres- 
pondence between every z-point and every w-poini, would 
have the junctions of its different sheets along the above- 
mentioned cuts. 


28. Joukowski's Aerofoil 
The transformation 

WKC lzc\* 

I I 4 t ^ \-*-J 

UJ+/CC \2+<V 

is important in the practical problem of mapping an 
aeroplane-wing profile on a nearly circular curve. If the 
profile has a sharp point at the trailing edge and we write 
jg = (2 JC)TT, then /? is the angle between the tangents to 
the upper and lower parts of the profile at this point. If a 
circle is drawn through the point c in the 2-plane, so that 
it just encloses the point z = c and cuts the line joining 
2 = c and 2 = c at 2 = c+* where is small, this circle 
is mapped by (1) on a wing-shaped curve in the w-plane. 

A special case of (1) when c = 1, K = 2 will now be 
discussed in detail. 

In practical problems on the study of the flow of air 
round an aerofoil, the transformation desired is one which 
maps the region outside the aerofoil on the region outside 
a circle or nearly circular curve. The special case of 
(1) when c = 1, *= 2, 

transforms a circle in the 2-plane, passing through the 
point z = 1 and containing the point 2 = 1, into a 
wing-shaped curve in the w?-plane, known as Joukowski's 

We readily see that (2) is the same as the trans- 
formation, already discussed, 

w = z + - . . . . (3) 

If C is a circle in the 2-plane passing through the point 
2 as 1, such that the point z = 1 is within (7, the trans- 
formation (3) maps the outside of C conformally on the 



outside of Joukowski's profile F. The shape of the curve F 
can easily be obtained from the circle G by making the 
point z trace out this circle, and adding the vectors z and 
1/z. See fig. 11. 

FIG. 11. 

We may also consider (2) as a combination of the three 




= *, 10: 

By the first of these, the circle C is transformed into a 
circle F in the -plane passing through t == 0. By the 
second, the circle F is transformed into a cardioid * in the 
-plane with cusp at = 0. The third transformation 
then maps the cardioid on the wing-shaped curve F in 
the w-plane. Since z = 1 corresponds to J = oo, the 
outside of C is mapped on the interior of F. The interior 
of the cardioid corresponds to the interior of F. Since 
= 1 corresponds to w = oo, the outside of F corresponds 
to the inside of the cardioid, and so to the outside of C. 

In fig. 11, C is the given circle, C' the circle obtained 
from C by the transformation l/z and Q is the unit circle. 

* See 21. 



The critical points of (3) are z = 1 and z = 1, and since 
the point z = 1 is inside C, the mapping of the outside of 
C on the outside of F is conformal. 

If the point z = 1 is outside (7, then the inside of F 
corresponds to the inside of (7, and the figure corresponding 
to this case is the same as fig. 11 with the circles G and C' 

29. Some Important Transformations Tabulated 

In Table 1 we tabulate for convenience a number of 
examples of domains in the 2-plane which can be mapped 
conformally on the interior of the unit circle | w | 


Domain in the z-plane 

Domain in the u?-plane 



Unit circle |f |<1 

Unit circle | w |^1 

iX *~ 

W ~ C dz-1 


Upper half -plane 

Unit circle | w\^.l 



Infinite strip of finite 
breadth oo^/^oo, 

Unit circle | w|^l 

w ~ l _ .iz 
w+l ** 


Area outside the ellipse 

Unit circle | u>|<l 



Area outside the para- 
bola r cos 2 J0 = 1 
Area within same para- 

Unit circle | w |^1 
Unit circle | u;|^l 
Unit circle | t0|^l 

w = tan 2 (i77\/2) 

w ~~~ i 

Some useful conformal transformations in which the 
domain in the to-plane is not a circle are given in Table 2. 
When the domain in the u>-plane is either the upper half- 
plane or a semicircle it can of course be transformed into 
a circle by either 2 or 7 of Table 1. 





> + 


1 > 

^> > I 

* 3 S S 




S S 




S o O 

< V A 

Upper half -plan 
Upper half -plan 








w |< 

cj a o 

2 ^ | 


1 A 


i * 

.2 II 





e angle 0<0<27r/n 
icircle | z \ < 1, t/> 


ii :A.S-XT; 
% J39^ 

- "So-s^-c 

"K o H I 

2 $ 3 : > 

5 O 



It is impossible, in the limited space at our disposal, 
to discuss all the transformations which are of practical 
importance. It is important, however, to mention briefly 
the Schwarz-Christoffel transformation, which has 
numerous important applications. 

Let a, 6, c, ... be n points on the real axis in the to-plane 
such that a<6<c<... ; and let a, jS, y, ... be interior 
angles of a simple closed polygon of n vertices so that 
a+j3+y+... = (n-2)7T. 

Then the transformation of Schwarz-Christoffel is a 
transformation from the w-plane to the 2-plane defined by 

dz ?! ft + y , 

~ ~ 

It transforms the real axis in the w-plane into the boundary 
of a closed polygon in the 2-plane in such a way that the 
vertices of the polygon correspond to the points a, 6, c, ... 
and the interior angles of the polygon are a, j8, y, .... 
When the polygon is simple, the interior is mapped by 
this transformation on the upper half of the to-plane. 
The number K is a constant, which may be complex. 

If we write K = Ae*\ where A and A are real, one 
vertex of the polygon can be made to correspond to the 
point at infinity on the to-axis. If a-* oo we can choose 


A to be of the form B(a)~" +l and since, as a-> oo, 

{(w a)/ a}^ -> 1, the transformation becomes 

dz > P . y_, 

t - l 

The reader who desires further information about this 
important transformation is referred to larger treatises.* 

* See e.g. Copson, Functions of a Complex Variable (Oxford, 
1935), p. 193 seq. 



1. Prove that, by the transformation 
c tza z+a 

w - 

two sets of coaxal circles are transformed into sets of confocal 
conies. What region of the t0-plane corresponds to the 
inside of the circle \(z a)/(z+a)| = J ? 

- -. + 
Z I 

the real axis in the z-plane corresponds to a cardioid in the 
to-plane. Indicate the region of the 3-plane which corres- 
ponds to the interior of the cardioid. 

3. If w = ic cot \z y where c is real, show that the rectangle 
bounded by x = 0, x = IT, y = 0, y = oo, is confonnally repre- 
sented on a quarter of the to-plane. Find a transformation 
=/(z) which maps this infinite rectangle on the semicircle 
||<a, >0. 

4. If w = tan z, prove that 

cot 2a?-l = 0, u'+v'-^v coth 2y+l = 0. 

Hence show that the strip %TT<X<%TT corresponds to the 
whole to-plane. To obtain a Riemann surface in the to-plane 
so as to secure unique correspondence between every t0-point 
and every z-point, show that the to-plane must be cut along 
the imaginary axis from i to oo and from t to oo. 
Investigate to = tan z as a combination of 

iw = (-!)/(+!), = 6 . 

6. Find the curves in the z-plane corresponding to | to | = 1 

w = 

6. Show that w = 2z/(l z 1 ) maps two of the four domains, 
into which the circles | z 1 | = V2, | z + l \ = V2 divide 
the 2-plane, confonnally on | w \ < 1. 

7. Prove that, if 3z a 2w + l = 0, the annulus 
l/\/3<| z \<l is mapped confonnally on the interior of the 


ellipse u*4-4y | = 4 cut along the real axis between its foci. 
Discuss what corresponds in the t0-plane to the curves 
(i) | z | = r, (ii) arg z = a. 

8. Find the transformation which maps the outside of the 
ellipse | s2 |+| z+2 \ = 100/7 on the circle | w |<1. 

9. Show that w = $(z + l/z) maps the upper half of the 
circle |z|<l on the upper half of the w-plane. At what 
points of the z-plane is the linear magnification equal to ? 
At what points is the rotation equal to 77 ? Prove that 
the magnification is greater than unity throughout the interior 
of the semicircle 3 | z | 2 = 1 in the upper half -plane. 

10. By considering the successive transformations 
= (z + l/z), w = l/ a prove that w = 4z 2 /(l+z 2 ) a maps 
the upper half of the circle | z |< 1 on the w-plane, cut along 
the positive real axis, so that the points z = 0, 1, i correspond 
to w = 0, 1, oo respectively. 

What points of the t0-plane correspond to z = 1 ? 

11. Show that by w = e 77 " 2 /* an equiangular spiral in the 
w -plane corresponds to a straight line in the z -plane. 

12. Discuss the transformation 

showing that the lines u = const., v = const, correspond to 
sets of confocal conies with foci at z = a, z = j8. 

13. Show that 2w = log{( l+z)/( lz)} represents |z|<l 
on the strip of the w-plane JTT<V< JTT. 

14. Show that iw = log{V(^/)~ 1} represents the strip 
v = 0, v = oo, u = TT, u = TT on the interior of the cardioid 
r = 2a(l+cos 0) in the z -plane cut along the real axis from 
the cusp to x = a. 

15. Show that, if c is real, 



conformally transforms the strip v= oo, t>= oo, w = 0, 
u s=s IT, into the circle | z \ ^ c. 

16. Show that, by the relation w 2 = l+e f , the linos 
x = const, are transformed into a series of confocal lemniscatea 
(Cassini's ovals) in the w-plane. 


If a>l and Z 2 (a 8 +t0* 1) = aw*, show that the interior 
of the circle | z \ = 1 is transformed into the interior of the 
Cassini's oval pp = a, where p and p are the distances of a 
point from the foci (1, 0) and ( 1, 0). 

17. If z = x+iy, prove that the inside of the parabola 
y* = 4c*(a;+c a ) is mapped on the upper half of the u>-plane by 


u? = t cosh 


18. Show that the transformation 

C ~ l-W 

transforms the inside of the circle | w \ = 1 with two semi- 
circular indentations, of centres 1 and 1, drawn so as to 
exclude these points from the circular area and boundary, 
into the annulus between two circles in the z -plane, of centre 
the origin and radii ce a , ce~, with a single slit along the real axis. 

19. If 0<a< 0<27T, show that 

w ^ (ze- 

maps the region a<argz</J on the K;-plane cut along the 
positive real axis. Hence find the transformation w =/(z) 
which maps the circular sector a<argz<0, |2J<1, on the 
circle | w |<1. 

20. Use the successive transformations 

C = (z+iM s = e*", t = - - , r = t\ w = -., 

18 r+i 

to form the single transformation w =/(z) which maps the 
strip <#<$, J/5^0 of the z-plane on | w |^1. 

21. Use the transformations = y% ^ = sin TT, w = - , 

to show that 

sin irr\/ 2 1 

W s - 


maps the inside of the parabola r = 2/(l -fcos 0) in the z-plane, 


out from the focus (z = 0) to the point z = oo, on the unit 
circle in the t0-plane cut from w = to w = 1. 

22. Find the equations of the curves in the s-plane which 
correspond to constant values of u and v if z = w+e w . What 
corresponds to the lines v = 0, v = ir ? Sketch some of 
the curves v = const, for values of v between TT and IT. 

23. Show that the transformation 

w/a = i sinh \(z ij8)/cosh (z+if)) 

transforms one of the regions bounded by the orthogonal 
circles | w \ = a and | w a cosec ft \ = a cot ft into the infinite 
strip 0<t/<j7T. 

24. If w = tanhz, show that the lines x = const., 
y = const., correspond to coaxal circles in the w-plane. 

Prove that this transformation maps the strip 
conformally on the upper-half of the w-plane. 

25. Prove that, if 0<c< 1, the transformation 




transforms the unit circle in the z -plane into the unit circle, 
taken twice, in the u;-plane, and the inside of the first circle 
into the inside, taken twice, of the second. 
26. Prove that, if a>0, 

maps the upper-half of the z-plane on the positive quadrant 
of the ttf-plane with a slit along the line v = TT, u^h 9 where 
w = h+in when z = I/a. (See p. 80.) 
27. Show that by the transformation 

dz __ 1 
dw ~~" 

the upper half of the w-plane can be mapped on the interior 
of a square, the length of a side of which is 


J n 



30. Complex Integration 

The development of the theory of functions of a 
complex variable follows quite a different line from that 
of functions of a real variable. In the latter theory, 
having discussed functions which possess a derivative, 
we proceed to consider the more special class of functions 
which possess derivatives of the second order ; then, 
from among those functions which possess derivatives of 
all orders, we select those which can be expanded in a 
power series by Taylor's theorem. In complex variable 
theory, on the other hand, we begin by dealing with 
regular functions, and, by virtue of the definition of 
regularity, the class of functions is so restricted that a 
function which is regular in a region possesses derivatives 
of all orders at every point of the region and the function 
can be expanded in a power series about any interior 
point of the region. 

By following Cauchy's development of complex variable 
theory, everything depends upon the complex integral 
calculus, and, in order to prove that a regular function 
possesses a second derivative, we must first of all express 
f(z) as a contour integral round any closed contour 
surrounding the point z. 

In order to develop the subject further we must now 
consider the definition of the integral of a function of a 
complex variable along a plane curve. 

The equations x = <f>(t), y = iff(t) 9 where a<J<j8, define 
the arc of a plane curve. If we subdivide the interval 



(a, jS) by the points a = * , t lt J 2 , ..., t r , ..., t n = j8, then 
the points on the curve corresponding to these values 
of t may be denoted by P , P l9 P 2 , ..., P n . The 
length of the polygonal line PoP v ..P n , measured by 

r~ yr-i) 2 }*> depends on the particular 


mode of subdivision of (a, jS). We call this summation 
the length of an inscribed polygon. If the arc be such that 
the lengths of all the inscribed polygons have a finite 
upper bound A, the curve is said to be rectifiable and A is 
the length of the curve. 

It can be shown that the necessary and sufficient 
condition that the arc should be rectifiable is that the 
functions <(), tfj(t) should be of bounded variation in (a, j3). 
If <f>'(t) and 0'() are continuous, it can be proved that the 
curve defined by x = <f>(t), y = *fj(t), a<tf<j8, is rectifiable 
and that its length s is given by * 

= f 


If we consider an arc of a Jordan curve whose equation 
is z = (j)(t)-{-iifj(t), where a^^jS, we define a regular 
arc of a Jordan curve to be one for which <f>'(t), */*'(t) are 
continuous in a^^jS. From the above theorems we 
see that the length of this regular Jordan arc is 

By a contour we mean a continuous Jordan curve 
consisting of a finite number of regular arcs. Clearly a 
contour is rectifiable. 

We now define the integral of a function of a complex 
variable z along a regular arc L defined by x = <f>(t), 

y = <!>(*), a<*< 

* For proofs and further details, see P.A., p. 205 acq., or G.I., 
p. 113. 


Let f(z) be any complex function of z, continuous along 
L y a regular arc with end-points A and J3, and write 
f(z) = u(x, y)+iv(x,y). Let z , z l9 ..., z n be points on 
L, z being A and z n being JB. Consider the sum 

(Zr-*r-l)}, . . . (1) 

where f is any point in the arc z r _ l9 z r . If f = r 4-^ f 
we write u f = u( r , 7j r ), v f = v(^ r , 7j r ), and (1) may be 

Now, by the mean-value theorem, 

X r X r _i =^(^)-^(^r-l) 

where ^ r _i<r r <^ f , ^ f -i<r f '<^ f . Hence the sum may be 

( Ur +iv r ){<j>'(r T ) +i^'(r r ')}(t r -t r _ l )]. . (2) 


Since all the functions concerned are continuous, and 
therefore uniformly continuous, we can, given e, find 8(e) 
so that 

for every r, provided that each | t r t r _ l |<8. Also 


It follows that, as and 8 tend to zero, 


tends to the same limit as 

{(* y,) 


that is to the limit 


Similarly the other terms of (2) tend to limits, and we find 
that the whole sum tends to the limit 


This limit (3) is taken as the definition of the complex 
integral of f(z) along the regular arc L, and it is written 



The integral of /(z) along a contour (7, consisting of a finite 
number of regular arcs L r , is given by 

f f(z)dz=Z f 

J C r J 


C r J L r 

31. An Upper Bound for a Contour Integral 

I. // f(z) is continuous on a contour L, of length I, on 
which it satisfies the inequality \ f(z) \ ^M, then 



It suffices to prove this theorem for a regular arc L. 
Since the modulus of any integral of a function of a 
real variable cannot exceed the integral of the modulus 
of that function, we have 

U/w* -|J 



= ML 

If C is a closed contour we make the convention that the 
positive sense of description of the contour is anti-clockwise. 


32. Cauchy's Theorem 

The elementary proof of Cauchy's theorem, which 
depends on the two-dimensional form of Green's theorem, 
requires the assumption of the continuity of f'(z). We 
first give a proof with this assumption, but, on account 
of the fundamental importance of Cauchy's theorem in 
complex variable theory, we shall also prove the theorem 
under less restrictive assumptions. 

II. The elementary proof of Cauchy's theorem. 
If f(z) is a regular function and if f'(z) is continuous 
at each point within and on a closed contour C f , then 

(2)dz = (1) 

Let D be the closed domain which consists of all points 
within and on C. Then by 30 (3) we can write the 
integral (1) as a combination of curvilinear integrals 

I f(z)dz= I (udx vdy)-\-i\ (vdx -\-udy). 
J c J c J c 

We transform each of these integrals by Green's theorem,* 
which states that, if P(x, y), Q(x, y), dQ/8x, 8P/dy are all 
continuous functions of x and y in D, then 

Since f'(z) = u x -\-iv x = v y iu v , and, by hypothesis, 
f'(z) is continuous in D, the conditions of Green's theorem 
are satisfied and so 

= o, 

by virtue of the Cauchy-Riemann equations. 
* See P.A., pp. 290-1, or G.I., p. 64 


It was first shown by Goursat that it is unnecessary to 
assume the continuity of f'(z) and that Cauchy's theorem 
holds if we only assume that/'(z) exists at all points within 
and on C. In fact the continuity of f'(z) 9 and indeed its 
differentiability, are consequences of Cauchy's theorem. 

Second proof of Cauchy's theorem. 

If f(z) is regular at all points within and on the closed 
contour C then 

f f(z)dz = 0. 
J c 

The integral certainly exists, for a regular function f(z) 
is continuous and a continuous function is integrable. 
We observe also that, if we construct a network of squares, 
by lines parallel to the axes of x and y, having the contour 
C as outer boundary, then C is divided into a network 
of meshes, either squares or parts of squares, such that 




where y denotes the boundary of a mesh described in the 
same sense as C. 

If z lies inside a square contour 8 of side a, then 


\z-z \ \dz\ 

<4v/2a 2 = 4V2(Area of S). 

This follows at once from 31, for | z z \<a\/2 and the 
length of the contour S is 4a. *^ 

We now prove two lemmas. 

Lemma 1. IfC is a closed contour, I dz = 0, zdz = 0. 

J c J c 

These results both follow from the definition of the 
integral, for 



dz = lim E {{z f z f _i).\} = 0, as max | z r z r _ l \-+ 0. 

7 -1 



zdz = lim 2{z r (z r z r _J} = lim {z r -i(z f z r _i)} 
J c 

= 0, 

Lemma 2. Goursat's Lemma. Given , then, by suitable 
transversals, we can divide the interior of C into a finite 
number of meshes, either complete squares or parts of 
squares, such that, within each mesh, there is a point z 
such that 

for all values of z in the mesh, where \ e y |<. 

Suppose the lemma is false ; then, however the interior 
of C is subdivided, there will be at least one mesh for 
which (1) is untrue. We shall show that this necessarily 
implies the existence of a point within or on C at which 
f(z) is not differentiate. 

Enclose C in a large square F 9 of area A, and apply 
the process of repeated quadrisection. When F is 
quadrisected there is at least one of the four quarters of 
F for which (1) does not hold. Let F l be the one chosen. 
Quadrisect F l9 choose one quarter of F 19 and so on. We 
thus obtain an unending sequence of squares F l9 F 2 , ..., 
r n , ..., each contained in the preceding, for which the 
lemma is untrue. These squares determine a limit-point f , 
and it is clear that must lie within C. 

Since f(z) is differentiate at , 

where, for sufficiently small values of [2 |, |^|<. 
Now all the F r , from one particular one onwards, lie within 
a circle, of centre , for which | z - | is so small that 
| f |<. This gives a contradiction, for by taking to 
be z , (1) is satisfied. This proves Goursat's lemma. 



Proof of the theorem. Some of the meshes y obtained 
by the subdivision of the interior of G will be squares, 
others will be irregular, since we are not concerned with 
the exterior of G. 

Integrate (1) round the boundary of each mesh. By 
virtue of lemma 1, we get 

f(z)dz = y | z Z Q | dz\ 

J Y J y 

and so, by addition, 

f(z)dz = H\ Y \z-z Q \dz, 

J C J Y 


If y is not a complete square, divide it into two parts, 
y x consisting of straight pieces, y a consisting of parts of (7. 
Since I >!<, 


A being the area of the large square F surrounding G. 
Also, the sum of the lengths of the portions y a cannot 
exceed the length I of the rectifiable curve (7, and so 

f eylz-zJ 
I J Y* 

where K is the length of the diagonal of JT, since | z Z Q 
We deduce that 



wtere B is a constant, and, since e is arbitrary, the theorem 
is proved. 


33. Cauchy's Integral, and the Derivatives of a 
Regular Function 

By means of Cauchy's integral we can express the 
value of a regular function /(z) at any point within a closed 
contour C as a contour integral round C. 

III. If f(z) is regular within and on a closed contour G 9 
and if be a point within (7, then 

Describe about z = a small circle y of radius S lying 
entirely within (7. In the region between C and y the 

FIG. 12. 

function (f>(z) = f(z)/(zt > ) is regular. By making a 
cross-cut joining any point of y to any point of C we 
form a closed contour F within which <}>(z) is regular, so 
that, by Cauchy's theorem, 

<l>(z)dz = 0. 

In traversing the contour F in the positive (counter- 
clockwise) sense, the cross-cut is traversed twice, once in 

aarh a^naa nnrl an if. fnllnwa f.hnf. 

each sense, and so it follows that 

Jo Jy 


J_ f , (z)dz I f f(*)dz = J_ f f(?)dz + JL 


Now on y, z J = 8e*0, and so the first of the two 
terms on the right becomes 

and, by theorem I, the modulus of second term on the 
right of (2) cannot exceed 

JL max |/(z)-/() | . 2778. 

Since /(z) is continuous at z = J this expression tends to 
zero as S-> : this proves the theorem. 

The next theorem shows how to find the value of 
/'() as a contour integral. 

IV. // /(z) is regular in a domain D, its derivative is 
given by 

o \ z ) 

where C is any simple closed contour in D surrounding the 
point z = J. 

We have, by III, 


If we now prove that | / |-> as | ^ [-> 0, the required 
result is established. Since /(z) is regular in and on 
it is bounded, so that |/(z) [^M on (7. Let d be the 


lower bound of the distance of from G : suppose h chosen 
so small that | h \<\d, then 

** 2n d*.$d 9 

where I is the length of (7. It is now clear that the term 
on the right tends to zero as | h |-> 0. 

V. // f(z) is regular in a domain D, then f(z) has, at 
every point of D, derivatives of all orders, their values 
being given by 

If we assume the theorem proved for = m and 
consider the expression 


we can readily prove that it is equal to 
(m+1)! f f(z)dz , 

i)i r 

" Jo 

2ni Jc(*-) m+a ' ' 

and the proof that | / | tends to zero as | h \~> follows the 
same lines as in IV. The details are left to the reader. 

34. Taylor's Theorem 

VI. If f(z) is regular in \ za |<p, and if is a point 
such that | a | = r(<p) then 


where a n =/ (n >(a)/n !. 

Let C be a circle of radius />', centre z = a, where 
r<p'<p, and consider the identity 

^- a (t-g) 1 ^ (g-a) n 

- a (2-a) 2 '" (z-a) 


Multiply each term by /(z)/27rt and integrate round 
we clearly obtain 

\n i;i 


This is Taylor's theorem with remainder. 
Since | /(z) | < Jf on (7 we readily see that 


where J? is a constant independent of n. Since r</>' we 
see that | R n |-> as n->oo. 


It therefore follows that the series 27a n (f a) n is con- 

vergent and has/(^) as its sum-function. If f(z) is regular 

in the whole z-plane, the expansion is valid for all . 

Corollary. If [/() | has a maximum M(r) on 
| a | = r</> <Aen, t/a n =/ (n) (a)/n !, w;e Aave <Ae inequality 

For, if C be the circle | za \ = r we have 

1 M(r) n M(r) 

35. The Theorems of Liouville and Laurent 

VII. Liouville's Theorem. // /(z) is regular in the 
whole z-plane and if |/(z) | <K for all values of z, then 
/(z) must be a constant. 

Let z l9 z 2 be any two points and a circle of centre z l 


and the radius p>2 |2 1 z 2 1, so that, when z is on (7, 
. By III, 

so that 

- * i r fa-sj/tt* < i r 

- 0^1 J c (z_ Zl)(2 _ 22 ) < 27 r J 

Keep 2, and z 2 fixed and make p->oo, then it follows that 
/(Zj) = /(z 2 ) ; in other words, /(z) is a constant. 

VIII. Laurent's Theorem . Let C l and C 2 be two circles of 
centre a with radii p l and p 2 (p 2 <Pi) 5 th en > tff( z ) be regular 
on the circles and within the annulus between (/j and <7 a , 

J being any point of the annuliis. The coefficients a n and 
b n are given by 

By making a cross-cut joining any point of O x to any 
point of C t , we readily see that 


Consider the two identities 

JL s J_j._iz^^. - (C-^)"- 1 , (C-) n i 

- z-a' t "(2-a) a " f "*"" r " ( 2 -o) "" (z-a)" *- 

z - 2 -""- 1 z - n i 



O^n it Mows that 


(z-a)(z--)' 27* 

Now P n is precisely the same remainder as in Taylor's 
theorem and we can prove, in the same way as in VI, 
that | P n |-> as 

Also we have 

f f^Y&^ 

Jc.l^-a/ -{ 

where r = | ^ a | and |/(z) 
it follows that | Q n |-> as U-+CQ 
If, therefore, we write 

' on (7 2 . Since /) a <r, 

where ^(f) = a n (-a) and /,() = MC-a)-, we 

o o 

see that/(^) converges forp 2 ^| ^ a |^/)j. 

It also follows that/ 1 (^) is regular and converges for 
| fl |^PI and that / a () is regular and converges for 

36. Zeros and Singularities 

If f(z) is regular within a given domain Z), we have seen 
that it can be expanded in a Taylor series about any 
point z = a of D and 


If a Q = a l = ... = a m-1 = 0, a m 4= 0, the first term in 
the Taylor expansion is a m (z a) m . In this case f(z) is 
said to have a zero of order m at z = a. 

A singularity of a function f(z) is a point at which 
the function ceases to be regular. 

If /(z) is regular within a domain D, except at the point 
z = a, which is an isolated singularity of /(z), then we can 
draw two concentric circles of centre a, both lying within 
D, The radius of the smaller circle p a may be as small 
as we please, and the radius p l of the larger circle of any 
length, subject to the restriction that the circle lies wholly 
within D. In the annulus between these two circles, 
/(z) has a Laurent expansion of the form 

/(z) = f a n (z-a)+f b n (z-a)~\ 
o i 

The second term on the right is called the principal part 
of /(z) at z = a. 

It may happen that b m * while 6 m+1 = 6 m + 2 ... = 0. 
In this case the principal part consists of the finite number 
of terms 

and the singularity at z = a is called a pole of order m 
of /(z) and the coefficient b l9 which may in certain cases 
be zero, is called the residue of /(z) at the pole z = a. 
If the pole be of order one, b l = lim{(z a)/(z)}. 

If the principal part is an infinite series, the singularity 
is an isolated essential singularity. 

(1) If z = a is a zero of order m of /(z), we now prove 
that this zero is isolated : in other words, there exists a 
neighbourhood of the point z = a which contains no other 
zero off(z). 

Clearly we can write /(z) = (z a) m ^(z), where ^(z) is 
regular in | a \<p and <j>(a) * 0, since </>(a) = a m . 


Write ^(a) =s 2c, then, since <f>(z) is continuous, there 
exists a region | z a |<8 in which |^(z) ^(a) | < | c \. 

where | za |<8, and so ^(z) does not vanish in | z a |<8. 

(2) If z = a is a pole of order m of /(z) it follows, from 
the definition of a pole by means of Laurent's theorem, 
that poles are isolated, for, the small circle, of centre z = a 
and radius p a , encloses the only singularity of /(z) within 
the domain D which contains the annulus between the 
two circles of radii p l and p a . 

(3) ///(z) has a pole at z = a, ften |/(z) [->oo a$ z-* a 
tn ant/ manner. For, if the pole be of order m, 

f(z) = (-a)--{6 m +6 m . 1 (z~o)+...+6 1 (-ar-Hra n (2~ar^} 


and, since 6 m * 0, we may write /(z) = (z a)- m ^r(z), 
where ^(z) is regular in | za [</>, and 0(a) = 6 m ( 4 s 0). 
Hence, by (1), we can find a neighbourhood | za |<8 
of the pole in which | ^r(z) |>J | 6 m |, from which it follows 

l/(*)l>ilMI*-*|- m . 

Hence |/(z) |-> co as z-> a in any manner. 

(4) Limit points of zeros and poles. 

Let a x , a 2 , ..., a n , ... be a sequence of zeros of a function 
/(z) which is regular in a domain D. Suppose that these 
zeros have a limit point a which is an interior point of D. 
Since /(z) is a continuous function, having zeros as near 
as we please to a, /(a) must be zero. Now z = a cannot 
be a zero of /(z), since we have proved in (1) that zeros 
are isolated. Hence /(z) must be identically zero. 

If /(z) is not identically zero in Z), then z = a must 
be a singularity of /(z). The singularity is isolated, but 
it is not a pole, since |/(z) | does not tend to infinity as 
z-+ a in any manner. Hence a limit point of zeros must 
be an isolated essential singularity of /(z). 


If f(z) be regular, except at a set of points which are 
singularities c l9 c a , ..., c n , ..., infinite in number, and 
having a limit point y in D, then y must be a singularity 
of/(z), since /(z) is unbounded in the neighbourhood of y. 
Since y is not isolated it cannot be a pole. We call such 
a singularity a non-isolated essential singularity. 

Examples, sin 1/z has an isolated essential singularity 
at z = 0. It is the limit point of the zeros, z = l/n?r, (n = J^l, 
2, ...). tan 1/2 has poles at the points z = 2/nrr, (n = 1, 
3, ...)> and so the limit point of the poles, z = 0, is a non- 
isolated essential singularity. 

Note on the region of convergence of a Taylor series. 
Iff(z) be a function which is regular, except at a number 
of isolated singularities at finite points of the z-plane, 

then we can expand f(z) in a Taylor series Sa n (za) n 

about any assigned point z = a, and the radius of con- 

vergence p of this power series will be the distance from 
z = a to the nearest singularity of/(z), since /(z) is clearly 
regular in | z a |<p, and cannot be regular in any circle 
of centre a whose radius exceeds p. 

We see that the radius of convergence of a power 
series depends upon the extent of the region within which 
the sum-function is regular, and it may be controlled by 
the existence of singularities which do not necessarily 
lie on the real axis. 

If we consider the real function 1/(1 a), the binomial 
expansion leads to 

the series being convergent if | x \<1 This seems quite 
natural, since the sum-function has a singularity at x = 1. 
However, on considering the function l/(l+x a ), we have 

..... (2) 


and the series is again convergent only if | x |<1 ; but 
if we regard l/(l+# 2 ) as a function of the real variable x 
there is nothing in the nature of the function to suggest 
the restriction of its range of convergence to |#|<1. 
If, however, we consider l/(l+z 2 ), where z is complex, 
the restriction on the region of convergence is at once 
evident, since l/(l+z 2 ) has singularities at z = , and 
the radius of convergence of the series (2), if a; is complex, 
is the distance of the origin from the nearest singularity 
and this is plainly unity. 

37. The Point at Infinity 

In complex variable theory we have seen that it is 
convenient to regard infinity as a single point. The 
behaviour of /(z) "at infinity " is considered by making 
the substitution z = l/ and examining /(!/) at = 0. 
We say that /(z) is regular, or has a simple pole, or has 
an essential singularity at infinity according as /(!/) 
has the corresponding property at = 0. 

We know that if /(!/) has a pole of order m at f = 0, 
near = we have 

n-O 2 b 

and so, near z = oo, 

/(z) = 2 a n z~+b l z+b 2 z*+...+b m z<. 


Thus, when /(z) has a pole of order m at infinity, the 
principal part of /(z) at infinity is the finite series in 
ascending powers of z. 


the function sinz has an isolated essential singularity at 
infinity, the principal part at infinity being an infinite series. 


38. Rational Functions 

Theorem. If a single-valued function f(z] lias no essential 
singularities either in the finite part of the plane or at infinity, 
thenf(z) is a rational function. 

Since the point at infinity is not an essential singularity 
of f(z), we can surround it by a region in which f(z) either 
is regular or has the point at infinity as its only singularity. 
That is, we can draw a circle (7, with centre the origin, 
such that the point at infinity is the only singularity 
outside G. There can only be a finite number of singularities 
within C, since poles are isolated singularities. Suppose 
that the poles inside G are at a v a 2 , ..., a n . The principal 
part at a f may be written 


z-a, ^ (z-a,) 2 ^ " ^ (z-a,)-' 

a, being supposed to be a pole of order m. The principal 
part at infinity is of the form 

Now consider the function 

The function ^(2) is plainly regular everywhere in the plane, 
even at infinity : hence <f>(z) is bounded for all z, and so, 
by Liouville's theorem, <f>(z) is a constant. Hence 

f(z) - 0+2 

and so /(z) is a rational function of 2. 

39. Analytic Continuation 

Suppose that / x (z) and / 2 (z) are functions regular in 
domains D l and J5, respectively and that D l and D 2 have 


a common part, throughout which f^z) = / a (z), then we 
regard the aggregate of values of f^(z) and / 2 (z) at points 
interior to D l or D 2 as a single regular function (f>(z). 
Thus <f>(z) is regular in A = D^-\-D^ and <f>(z) = / 1 (z) in 
JDj and <f>(z) = / 2 (z) in Z) 2 . The function / 2 (z) may be 
regarded as extending the domain in which f^z) is defined 
and it is called an analytic continuation of /^z). 

The standard method of continuation is the method of 
power series which we now briefly describe. 

Let P be the point z , in the neighbourhood of which 
f(z) is regular, then, by Taylor's theorem, we can expand 
f(z) in a series of ascending powers of z z , the coefficients 
in which involve the successive derivatives of f(z) at z . 
If S be the singularity of f(z) which is nearest to P, then 
the Taylor expansion is valid within a circle of centre P 
and radius PS. Now choose any point P l within the 
circle of convergence not on the line PS. We can find 
the values of f(z) and all its derivatives at P l9 from the 
series, by repeated term-by-term differentiation, and so 
we can form the Taylor series for f(z) with P l as origin, 
and this series will define a function regular in some circle 
of centre P v . This circle will extend as far as the singularity, 
of the function defined by the new series, which is nearest 
to P l and this may or may not be S. In either case the 
new circle of convergence may lie partly outside the old 
circle and, for points in the region included in the new 
circle but not in the old, the new series may be used to 
define the values of f(z) although the old series failed to 
do so. 

Similarly, we may take any other point P 2 in the region 
for which the values of the function are now known and 
form the Taylor series with P 2 as origin which will, in 
general, still further extend the region of definition of the 
function ; and so on. 

By means of this process of continuation, starting from 
a representation of a function by any one power series, 
we can find any number of other power series, which 


between them define the value of the function at all points 
of a domain, any point of which can be reached from P 
without passing through a singularity of the function. 
It can be proved that continuation by two different paths 
PQR, PQ'R gives the same final power series provided that 
the function has no singularity inside the closed curve 

We may now, following Weierstrass, define an analytic 
function of z as one power series together with all the 
other power series which can be derived from it by analytic 
continuation. Two different analytic expressions then 
define the same function if they represent power series 
derivable from each other by continuation. The complete 
analytic function defined in this way need not be a one- 
valued function. Each of the continuations is called an 
element of the analytic function. 

If f(z) is not an integral function there will be certain 
exceptional points which do not lie in any of the domains 
into which /(z) has been continued. These points are 
the singularities of the analytic function. Clearly the 
singular points of a one- valued function are also singularities 
in this wider sense. 

There must be at least one singularity of the analytic 
function on the circle of convergence (7 of the power 


series a n (z z ) w .* For, if not, we could construct, by 


continuation, a function equal to /(z) within <7 but regular 
in a larger concentric circle jT . The expansion of this 
function in a Taylor series in powers of z z would then 
converge everywhere within F Q . This is impossible, since 
the series would be the original series whose circle of 
convergence is (7 . If z l is any point within <7 , let C l 
be the circle of convergence of the power series 

* For a proof of this, and further details, see Titchmarsh, 
Theory of Function* (Oxford, 1932), p. 145. 


If Q l is the circle of centre z l which touches (7 internally, 
the new power series is certainly convergent within Q l 
and has the sum /(z) there. There are now three 
possibilities. Since the radius of C l cannot be less than 
that of Q v we have either (i) C l has a larger radius than 
Q l9 or (ii) (7 is a natural boundary * of /(z), or (iii) C l may 
touch (7 internally, though (7 is not a natural boundary 


In case (i) C l lies partly outside (7 and the new power 
series provides an analytic continuation of /(z) : we can 
then take a point z a within C l and outside C and repeat 
the process. In case (ii) we cannot continue /(z) outside 
(70 and the circle C l touches (7 internally no matter what 
point z l within (7 is chosen. In case (iii) the point of 
contact of (70 and G l is a singularity of the analytic function 
obtained by continuation of the original power series. 
For there is necessarily one singularity on C l and this 
cannot be within (7 . 

We may illustrate some of the above remarks by the 
following examples. 

1. The series 

represents the function f(z) = l/(az) only for points within 
the circle \z\ =* | a |. If 6/a is not real, the series 

1 *-b (s-6) 

a-6 + (a-6) 1 "*" (a~6) "*" "' f 

for different values of 6, represents f(z) at points outside the 
circle | z \ = | a |. 

2. That there are functions to which the process of 
continuation cannot be applied may be seen by considering 
the function 

g(z) = l+2 2 +z* + ...+* |n + .... 

* See Example 2 below. 


It is readily shown that any root of any of the equations 

2 1 1 2 4 1 * 1 ! I 

Z - 1, Z - 1, 1, 9 ... 9 

is a singularity of g(z) 9 and hence that on any arc, however 
small, of the circle | z \ = 1 there is an unlimited number 
of them. The circle | z \ = 1 is in this case a natural boundary 
of g(z). This illustrates case (ii) above, 

40. Poles and Zeros of Meromorphic Functions 

A function /(z), whose only singularities in the finite 
part of the plane are poles, is called a meromorphic 
function. We now prove a very useful theorem. 

// /(z) is meromorphic inside a closed contour C, and 
is not zero at any point on the contour, then 

*- - <" 

where N is the number of zeros and P the number of poles 
inside 0. (A pole or zero of order m must be counted m 

Suppose that z = a is a zero of order w, then, in the 
neighbourhood of this point 

/(z) = (z-a)^(z), 
where <f>(z) is regular and not zero. Hence 


/(z) z-a "" ftz) ' 

Since the last term is regular at z = a, we see that /'(z)//(z) 
has a simple pole at z = a with residue m. Similarly, if 
z = 6 is a pole of order k, we see that /'(z)//(z) has a simple 
pole at z = 6 with residue ft. It follows, by 33, III, 
that the left-hand side of (1) is equal to ZmSk = NP. 
If /(z) is regular in (7, then P = 0, and the integral 
on the left of (1) is equal to N. Since 


we may write the result in another form, 

W) dz==Al 

where A c denotes the variation of log/(z) round the 
contour C. The value of the logarithm with which we 
start is immaterial ; and, since 

and log |/(z)| w one- valued, the formula may be written 
N=-A *Tgf(z). 

This result is known as the principle of the argument. 

41. Rouch6 's Theorem 

If f(z) and g(z) are regular within and on a closed contour 
C and | g(z) \ < \ f(z) \ on C, then f(z) and f(z)+g(z) have 
the same number of zeros inside C. 

We observe that neither f(z) nor f(z) +g(z) has a zero 
on (7, and so, if N is the number of zeros of f(z) and N' 
the number of zeros of f(z)+g(z), 

2<jrN = A c arg/, 

27rN' = AC arg (f+g) = A Q arg /+ A arg ( J + y) 
The theorem is proved if we show that 

+j\ = 0. 

Since | g \ < |/|, the point w = l+g/f is always an 
interior point of the circle of centre w = 1 and radius 
unity : thus, if w = pety, <f> always lies between \n 
and \n and so arg (1 +g/f) = ^ returns to its original value 
when z describes C. Since <f> cannot increase or decrease 
by a multiple of 2rr, the theorem follows. 


The preceding theorems are useful for locating the 
roots of equations. The method is illustrated by the 
following example. 

Example. Prove that one root of the equation z 4 +z 8 + l =0 
lies in the first quadrant. 

The equation z 4 +z 3 + l = plainly has no real roots. 
For, if we put z = x, x*+x* + l = has no real positive 
roots. If we put z = a; and write $(x) = x* # 3 + 1=0, 
we see that 

, if x>\ ; 
and <f>(x) =x*+(l~x)(x*+x + l)>0 9 if 

Hence the given equation has no real negative roots. 

The given equation has no purely imaginary roots either, 
for, on putting z = iy, we get y 1 iy* + l = and it is plain 
that the real and imaginary parts never vanish together. 

Consider A arg (z 4 +z 8 + l) round part of the first quadrant 
bounded by | z \ = R where R is large. On the arc of the 
circle, z = ReiO, and we have 

A arg ( 2 4+z 8 + l) = j arg (R* e *iO)+ A arg {1 +O(R~ 1 )}, 
= 2<jr 

On the axis of y we have 

arg(z 4 +z s + l) = arc tan 

~~ y 

The numerator of y*/(y l + l) only vanishes when y = and 
the denominator does not vanish for any real y. Hence as 
y ranges from oo to along the imaginary axis, the initial 
and final values of arg (z 4 +z a + l) are zero. Hence the total 
change in arg(z 4 +z 8 + l), where R is large, is 2w. It follows 
that one root of the given equation lies in the first quadrant. 

42. The Maximum-Modulus Principle 

We now establish an important theorem which may 
be stated as follows. 

// /(z) is regular within and on a dosed contour C 9 then 
|/(z)| attains its maximum value on the boundary of G 
and not at any interior point. 


Lemma. If<f>(x) is continuous, (f>(x)^,K and 

. (1) 

then (f>(x) = K. 

Suppose that ^(x 1 )</c, then there is an interval 
(x l 8, Zx+S) in which </>(X)^.K and 

which contradicts (1). 

Theorem. If |/(z)|<Jf on C, then \f(z)\<M at all 
interior points of the domain D enclosed by C, unless f(z) 
is a constant j in which case \f(z)\ = M everywhere. 

Suppose that at an interior point z of D, \f(z)\ has 
a value at least equal to its value elsewhere. Let F be 
a circle of centre Z Q lying entirely within D. Then by 

33, in, 

/w _ ' r /w* . . . (2) 

2<TTl J r Z-Z Q 

Write z z = re^, f(z)/f(z Q ) = petf, so that p and ^ are 
functions of 0, then (2) may be written 

1 f2 


^ J o 

MM. . . . (3) 
Hence 1 < ^- I odd. 

By hypothesis p^l, and so, by the lemma, p = 1 for all 
values of 0. On taking the real part of (3) we get 


and so, by the lemma, cos^i = 1. Hence f(z) =/(z ) a 
JT. Since f(z) is a constant at any point a on J*, it follows 
by Taylor's theorem that it is constant in a neighbourhood 
of a, and hence, by analytic continuation, f(z) is constant 
everywhere within and on C. 

There is a corresponding theorem for harmonic functions. 
A function which is harmonic in a region cannot have a 
maximum at an interior point of the region. 


1. The function /(z) is regular in | z a \<R ; prove that, 
if 0<r<R, 

I /-27T .- 

f'(a) - _ P(e)e-* dO, 

7TT J Q 

where P(0) is the real part off(a+reiO). 

2. <f>(z) and $(z) are two regular functions ; z = a is a 
once repeated root of $(z) = and <f>(a) ^ 0. Prove that 
the residue of <t>(z)l\ff(z) at z = a is 

{6 ^'(a)0"(a) -2 #a)0'"(a)}/3 {f' (a)}. 

3. The function f(z) is regular when | z \<R'. Prove 
that, if | a \<K<R', 

where O is the circle | z \ = B. Deduce Poisson's formula 
that, if 0<r<K, 

1 T 27r # a r* 

T, / *-2Kr C o* ( e-t )+ r* 

4. By using the integral representation of / (n) (a), ( 33, V), 
prove that 

ajfl \ t _ 1 f x " e *' 
fTl) = 2^ J c nTz*+* *' 

where C is any closed contour surrounding the origin. Hence 
prove that 


6. Obtain the expansion 


- ... 

2 J \ 2 / + 2>.3t / \2/^2.6! / \2/^ J 

and determine its range of validity. 

6. If f(z) = S z a /(4+n 2 z a ), show that f(z) is finite and 


continuous for all real values of z but/(z) cannot be expanded 
in a Maclaurin series. Show that /(z) possesses Laurent 
expansions valid in a succession of ring spaces. 

7. Prove that cosh (z+-j = a + a n (z n H ) , where 

\ zl i \ z n l 

1 r 27r 

o n == cos n6 cosh (2 cos 8)dd. 
2n J 

8. Find the Taylor and Laurent series which represent 
the function (z*-l)/{(z+2)(z+3)} in (i) | z |<2, (ii) 2<| 2 |<3, 

9. Find the nature and location of the singularities of the 
function l/{z(e f 1)}. Show that, if 0< | z |<27r, the function 
can be expanded in the form 

and find the values of a and a a . 

10. The only singularities of a single-valued function 
f(z) are poles of orders 1 and 2 at z = 1 and z = 2, with 
residues at these poles 1 and 2 respectively. If /(O) = 7/4, 
/(I) = 6/2, determine the function and expand it in a Laurent 
series valid in 1< | z |< 2. 

11. Classify the points z = 0, z = 1 and the point at 
infinity, in relation to the function 

/(,).!=? sin jij, 

and find the residues of /(z) at z = and at z = 1. 

12. Show that, if 6 is real, the series 


is an analytic continuation of the function defined by the 

13. The power series z + z 2 +i^ 8 + and 

have no common region of convergence : prove that they 
are nevertheless analytic continuations of the same function. 

14. If a>e, use Rouch6's theorem to prove that e* = az* 
has n roots inside the circle | z \ = 1. 

15. The Fundamental Theorem of Algebra. By taking 
f(z) = a z m , g(z) = c^z- 1 +a^z m ~ z + ... +a m , use RoucWs 
theorem to prove that the polynomial 

F(z) = a z"+a 1 z'- 1 + ... +a m 

has exactly m zeros within the circle \z\ = R for sufficiently 
large R. 

Deduce from Liouville's theorem that F(z) has at least 
one zero. 

16. Prove that 3 8 +3z 8 +7z+6 has exactly two zeros in 
the first quadrant. 

17. If |/(z) \>m on | z \ = a, /(z) is regular for | z \^a 
and |/(0) | <ra, prove that /(z) has at least one zero in | z | <a. 
(See 42.) 

Deduce that every algebraic equation has a root. (This 
is another proof of the Fundamental Theorem of Algebra.) 

18. If a domain D of the z -plane is bounded by a simple 
closed contour C and w =/(z) is regular in Daiid on C7, prove 
that, if /(z) takes no value more than once on C, then f(z) 
takes no value more than once in D. (Use the theorem 
of 40.) 

Prove that the above result holds for the function 
w = z a +2z+3, if D is the domain | z \ < 1 and C is the unit- 



43. The Residue Theorem 

We now turn our attention to the residue theorem, 
and to one of the first applications which Cauchy made 
of this theorem the evaluation of definite integrals. It 
should be observed that a definite integral which can be 
evaluated by Cauchy's method of residues can also be 
evaluated by other means, though usually not so easily.* 

We have already defined the residue of a function 
f(z) at the pole z = a to be the coefficient of (z a)~ l in 
the Laurent expansion of /(z), which, if z = a is a pole 
of order m, takes the form 

Za n (z-a)+3b n (z-a)-*. 
o i 

We have also remarked that, when z = a is a pole of 
order one, the residue b l can be calculated as lim{(z a)/(z)}. 


The residue can also be defined as follows. If the point 
2 = a is the only singularity of /(z) inside a closed contour 

1 f 

C y and if . I f(z)dz has a value, that value is the residue 
1m J Q 

of /(z) at z = a. 

The residue of /(z) at infinity may also be defined. 
If /(z) has an isolated singularity at infinity, or is regular 

e~ m * dx, sometimes stated to be an integral which 

cannot be evaluated by Cauchy's method, see Courant, Differential 
and Integral Calculus, II, p. 661. In this case Cauchy's method 
is the more difficult. 


there, and if G is a large circle which encloses all the finite 
singularities of/(z), then the residue at z = oo is defined to be 

taken round O in the negative sense (negative with respect 
to the origin), provided that this integral has a definite 
value. If we apply the transformation z = l/ to the 
integral, it becomes 

taken positively round a small circle, centre the origin. 
It follows that if 

has a definite value, that value is the residue of /(z) at 

Note that a function may be regular at z = oo but yet 
have a residue there. 

The function /(z) = A/z has a residue A at z = and a 
residue A at z = oo, al though /(z) is regular at z = oo. 

Theorem 1. Cauchy's Residue Theorem. 

Let f(z) be continuous within and on a closed contour 
and regular, save for a finite number of poles, within G. 

I f(z)dz = 2mSy%> 
J c 

where Z72 is the sum of the residues off(z) at its poles within C. 
Let a lf a 2 , ..., a n be the n poles within C. Draw a 
set of circles y r of radius 8 and centre a r , which do not 
intersect and which all lie inside O. Then /(z) is certainly 
regular in the region between C and these small circles y f . 



We can therefore deform C until it consists of the small 
circles y r and a polygon P which joins together the small 
circles as illustrated in fig. 13. 

FIG. 13. 


f f( z )dz = f f(z)dz+Z ( f(z)dz = 2 f f(z)dz, 

J C J P fl J 7r f = l J Yr 

for the integral round the polygon P vanishes because 
f(z) is regular within and on P. 
If a f is a pole of order m, then 

where <f>(z) is regular within and on y f . Hence 

f m C b 

f(z)d*=Z 7-~T^Z. 

J y r 8-1 J y r \ z a r) 

On writing z = a r -\-&eW, varies from to 77 as the point z 
makes a circuit of the circle y r , and so 

f f( z )dz = Zb 8 &-> \ * e(i-'Wid0 = 2irib v 
J Yr i J 

Hence j f(z)dz = Z \ f(z)dz = 2iri 

J C f-l J Yr 

which proves the theorem. 


Theorem 2. // lim{(z a)f(z)} = 6, and if C is the 


arc, #!< arg (2 a) <0 2 , o/ fe circle \ za \ = r, 
lim f /(z)dz 


Given c, we can find an 77 (e) such that, if | za |<ij, 
| S |<, where (z-a)/(z) = fr+8. 


f // 1 J f 6 + 8 J f *' 

/( 2 ) <fe = dz = 

Jo Jcz o Jfli 

i r 


and so, on taking the limit as r-> 0, the theorem follows. 

If 2 = a is a simple pole of /(z), 6 is the residue of f(z) 
at z = a, so that if the contour is a small circle surrounding 
the pole, a #1 == 2ir and we get 


/(Z) dz = 27N&. 

44. Integration round the Unit Circle 

We consider first the evaluation by contour integration 
of integrals of the type 


, sin0)d0, 

where <(cos0, sin#) is a rational function of sin0 and 
cos 0. If we write z = e*0, then 

f27T f 

and so I ^(cos 0, sin 0)d0 = 4t(z)dz, 

Jo Jo 


where 0(z) is a rational function of z and is the unit circle 
I z I = 1. Hence 


where Z7c denotes the sum of the residues of ^r(z) at its 
poles inside C. 

Example. Prove that, if a > 6 > 0, 

J = \ 

Now on making the above change of variable, if C is the 
unit circle | z \ = 1, 

j = JL J _XI l^ZI = J_ I _JL: i^I_ = 1 f ^( Z )^, 


are the roots of the quadratic z*+2az/b + l = 0. Since the 
product of the roots a, j8 is unity, we have | a || ]8 | = 1 where 
| ft \>\ a |, and so z = a is the only simple pole inside C. 
The origin is a pole of order two. We calculate the residues 
at (i) z = a, and (ii) z = 0. 


(i) Residue = lim(z-a)F(z) = lira ~ ^-' = - - ^ 
z-+a z^a z ( z -P> a ~P 

(z I) 1 
(ii) Residue is the coefficient of 1/z in - - - , 

Z \Z "T~iC**y(/ "|~ L) 

where z is small. Now 

z\z* + 2az/b + 1 ) 
and coefficient of l/z is plainly 2a/6. 


which proves the result. 

45. Evaluation of a Type of Infinite Integral 

Let Q(z) be a function of z satisfying the conditions : 
(i) Q(z) is meromorphic in the upper half-plane ; 
(ii) Q(z) has no poles on the real axis ; 
(iii) zQ(z)-+ uniformly, as | z |->oo, for 0<arg 


f [0 

iv) I Q(x)dx and Q(x)dx both converge ; then 
Jo J oo 

where EJ& denotes the sum of the residues of Q(z) at its 
poles in the upper half-plane. 

Choose as contour a semicircle, centre the origin and 
radius R, in the upper half-plane. Let the semicircle be 
denoted by I 7 , and choose R large enough for the semicircle 
to include all the poles of Q(z). Then, by the residue 

f Q(x)dx+ f Q(z)dz = 
J -R J r 

From (iii), if R be large enough, | zQ(z) |< for all points 
on JT, and so 

I f Q(z)dz = I { V Q(Ref8)R&OidO j <c f" d0 = TIC. 

Hence, as jR->oo, the integral round JT tends to zero. 
If (iv) is satisfied, it follows that 

/)//M\/7/M _ OUM|* ^*f&^ 

\\X](LX = ^TTl^i//^^. 

If Q(z) be a rational function of z, it will be the ratio 
of two polynomials N(z)/D(z), and condition (iv) is satisfied 


if the degree of D(z) exceed that of N(z) by at least two, 
for, when x is large, Q(x) behaves like x~* 9 where p^2 and 

f dx A f ~ M dx i. *u * * 
I and I both exist.* 

J A x> J -*x* 

Note that condition (iv) is required as well as (iii), for the 
condition xQ(x)->0 is not in itself sufficient to secure the 


convergence of I Q(x)dx. This can be seen by taking 
Q(x) = (* log *)-*. 

Example. Prove that, if a>0, 

If 2 4 +a* = 0, we have z 4 == a 4 e 7ri , and the simple poles 
of the integrand are at ae 7 ^/ 4 , ae 377 */ 4 , ae 57r ^/ 4 , ae 77ri / 4 . Of these, 
only the first two are in the upper half-plane. The conditions 
of the theorem are plainly satisfied, and so 

= 2iri E {Residues at z = ae 77 */ 4 , ae 37r */ 4 }. 

Let k denote any one of these, then A 4 = a 4 and the residue 
at the simple pole z = k is lim{(2 fc)(z 4 & 4 )- 1 }. This may 


be evaluated by Cauchy's formula, as applied to the evaluation 
of limits of expressions of the indeterminate form f 0/0, and so 



* For the convergence of infinite integrals, see P. A., p. 193, 
or G.I., p. 77. 
t P.A., p. 106. 


00 dx 

f ax 

Hence I . = 

j x*+a 4 

The theorem can be extended to the case in which 
D(z) = has non-repeated real roots, so that Q(z) has 
simple poles on the real axis. We now indent the contour 
by making small semicircles in the upper half-plane to 
cut out the simple poles on the real axis. Suppose that 
D(z) = has only one root z = a, where a is real. The 
contour is then as shown in fig. 14. The small semicircle 

is denoted by y, its centre is the point x = a and its 
(small) radius is />. If F is large enough to enclose all 
the poles of Q(z) in the upper half-plane, then the integral 
round F tends to zero as R-+OO, as before. We therefore 
have, if the path of integration be as indicated by the 
arrows in fig 14, 


f + P~ P + f + f* 
J T J -R Jy Ja 

fa-p [R f 

As IZ^oo, + Q(x)dx = P Q(x)dx, 

J -R J a+p J -oo 

and it remains to consider I Q(z)dz. Now, on y f 

z = a+/>e l *0, and so 

f Q(z)dz= t 
J y J 



Since Q(z) contains the factor (2 a)- 1 we may write 
Q(z) = <f)(z)I(za) and ^(z) is regular at and near z = a. 

f Q(z)dz = f <(>(a+peiO)id6 = t f 

J y J 7T J 

since <f>(a+peiO) is regular at and near a and can be expanded 
by Taylor's theorem with remainder when n = 1. It 
follows that 

Q(z)dz-+ 7n^(a) as p-> 0. 

Since ^(a) is plainly the residue of Q(z) = <f>(z)/(za) at 
z = a, we can write the final result in the form 


J -0 


where Z7 denotes the sum of the residues of Q(z) at its 
simple poles on the real axis, for clearly each pole on the 
real axis can be treated in the same way as z = a. The 
principal value of the integral is involved, because equal 
spaces p are taken on either side of the real poles, and, 
by definition,* 

lim (~~ P + ( ft f(x)dx = P ( ft f(x)dx. 

p-+Q J a J a+p J a 

It should be noticed that if a pole be cut out by a small 
semicircle, the contribution to the value of the integral is 
half what it would be if a small circle surrounded the pole. 
(See 43, Theorem 2.) 

46. Evaluation of Infinite Integrals by Jordan 's 

We now prove a very useful theorem which is usually 
known as Jordan's lemma. 

* See P.A., p. 195, or G.I., p. 81. 


If F be a semicircle, centre the origin and radius R, and 
f(z) be subject to the conditions : 

(i) f(z) is meromorphic in the upper half -plane, 
(ii) /(z)-> uniformly as \ z |-*oo/or 0<arg 
(ill) m is positive ; then 


e mit f(z)dz-+ as J?->oo. 

By (ii), if R is sufficiently large, we have, for all points 
on/ 1 , | /(z)|<. Now 

| exp miz \ = | exp{wiJ?(cos0+t sin0)}| = 

j f(z)e*dz = I f(z)e>i*ReiOid6 < J e-Ri 


2Re * 

Now it can be proved, by considering the sign of its 
derivative, or otherwise, that sinO/d steadily decreases 
from 1 to 2/7T as 6 increases from to Jw. Hence, if 


I f /(zjgm&dz < 2Re ( e-2M/nd0, 

m m 

from which the lemma follows. 

By using this lemma, we can evaluate another type 
of integral. The method may be set out as a theorem 
as follows. 


Let Q(z) = N(z)/D(z), where N(z) and D(z) are poly* 
nomials, and D(z) = has no real roots, then if 

(i) the degree of D(z) exceeds that of N(z) by at least one, 
(ii) m>0, 


Q(x)e mi *dx = 


where 27S+ denotes the sum of the residues of Q(z)e mi9 at 
its poles in the upper half -plane. 

If we write f(z) = Q(z)e m< * , we see that f(z) satisfies the 

conditions of Jordan's lemma and so I f(z)dz-> as jR->oo. 

On using the same contour as before, a large semicircle 
in the upper half-plane, by making B-+CQ we get 

Q(x)e mix dx = ^mETe*. 


On taking real and imaginary parts of this result we see 
that by this method we can evaluate integrals of the type 

I f(x) cos mx dx , f(x) sin mx dx. 

J -00 J -00 

By a well-known test for convergence of infinite in- 


tegrals,* if f(x) decreasesand -> as s->oo, since , c ? 8 mx dx 

J a sm 

is bounded, the integrals in question converge. 

Example. Prove that, if a>0, m>0, 
00 cosma;^ ir 


The only pole of the integrand considered, e 

in the upper half-plane, is a double pole at z = ai. The 

* The test is known as Dirichlet's test. See Titchmarsh, 
Theory of Functions (Oxford, 1932), p. 21. 


conditions of the theorem are easily seen to be satisfied 
and so 

dx =s 27rt{Residue of at z = ai}. 

jo (a 2 +a; 2 ) a (a 2 +z 2 ) a 

Put z = ai+t then, since t is small, 

gtnif e~ mo 6 m ** 6""*** 

(a a +z a ) a < 2 (2at+0 1 4a 2 * 2 

and the residue, which is the coefficient of tr l , is easily seen 
to be 

te- ma (l+ma) 

f e m<a! dx 7re~ tno (l+ma) 

Hence = ^ ' . 

J -oo (a 2 +x 2 ) 2 2a 8 

On equating real parts, and taking one half of the result 
for the integral from to oo, we get 

r cos mx dx TJ 

If there are simple poles of the integrand on the real 
axis, we get a modification of this result, similar to that 
obtained for the theorem of 45. Thus, if D(z) = has 
non-repeated real roots, we get 

r 00 

> I 0(x 

)e mi *dx = 

the proof following the same lines as before. 

An example of this extension of the theorem is the proof 

00 * 

! dx = JTT, if W>0. 

On considering the integrand e mi '/z we see that it has 
a simple pole at z = and none in the upper half -plane. 
The residue at z = is easily seen to be unity, and so we get 

r oo 0m< 

j 1 

J -at X 


On equating real and imaginary parts we get 

cos mx _ 
dx = 0, 


sin mx _ 
dx = ir i 



the " P '* is not necessary in the second integral, since 
sin mx/x^- m as #-> 0, whereas the integrand in the first integral 
becomes infinite at the origin. From the second result we get 

00 sin mx , 

dx = JTT. 


47. Integrals Involving Many- Valued Functions 

A type of integral of the form &- l Q(x)dx, where a 


is not an integer, can also be evaluated by contour 
integration, but since z a ~ l is a many-valued function, it 
becomes necessary to use the cut plane. One method of 
dealing with integrals of this type is to use as contour a 
large circle F, centre the origin, and radius B ; but we 
must cut the plane along the real axis from to oo and 
also enclose the branch-point z = in a small circle y of 
radius p. The contour is illustrated in fig. 15. 

Fio. 15. 


Let Q(x) be a rational function of x with no poles on 
the real axis. If we write f(z) = z?~ l Q(z) and suppose 
that z/(z)-> uniformly both as | z \ -> and as | z \ ->oo, 
then we get the integral round F tending to zero as 
jR->oo and the integral round y tending to zero as />-> ; 
for, on F, if -R is large enough, | zf(z) \ <e and so \f(z) \ </R 9 

Similarly on y, | z/(z)|< if p is small enough, and so 

f(z)dz < - 2np = 27T. 

Hence on making p-+ and R-+CQ we get 

/> fo 

&~ l Q(x)dx + 

JO Jo 

where Sy\* is sum of residues of /(z) inside the contour. 
We observe that the values of z*" 1 at points on the upper 
and lower edges of the cut are not the same, for, if z = re^, 
we have z*"" 1 = r*- V^*" 1 * and the values of z at points 
on the upper edge correspond to | z | = r, = 0, and at 
points on the lower edge they correspond to | z \ = r , 

= 277. 

Since e^wifa"" 1 ) - ^Trtc^ \^e get 



We also observe that, when calculating the residues at 
the poles, z a ~~ l must be given its correct value r a 
at each pole. 

Example. Prove that 

f &- l dx ir .. _ 

-7- = -: , if 0<o<l. 

Jo 1 +a? sin aw 


Here we observe that, when/(z) = z?~ l (l +z)~ l , z/(z)-> aa 
| z | ->oo, if a< 1, and z/(z)-> as | z \ -> 0, if a>0. Hence, if 
0<a<l, the integral round F tends to zero as .R->oo and 
the integral round y tends to zero as p-> 0. 



J o 

{Residue of z- l (l+z)~ l at z = -1}. 

1+3? 1- 

At z = 1 we have r = 1, $ = w , and so, for the residue, 

lim f , % z - 1 ) , , . . 

g -*-l 1 (1+z) f = ( I)*"" 1 = e (a ~ 1)7r< = 


f * &~ l , . f e 1 " 7 ' 

Jo 1+5* = ~ 27rt I !=* 

Bin air 

This integral can also be evaluated by integrating 
) t using as contour a large semicircle in the upper 
half plane and the real axis indented by semicircles at 
2 = and at z = 1. In this case the cut plane is 
unnecessary. The evaluation of the integral by this 
second method is left as an exercise for the reader.* By 
this second method one obtains the further result that 



Jo 1-* 


48. Use of Contour Integration for deducing 
Integrals from Known Integrals 

The contours used so far have been either circles or 
semicircles, and although a large semicircle in the upper 
half plane is generally used for integrals of the type 
discussed in 45, there is no special merit in a semicircle. 
The rectangle with vertices <R, 7J+iJ? could also be 

* Bee Copaon, Functions of a Complex Variable, p. 140. 



used in these cases. We now give two examples of 
deducing the values of some useful integrals by integrating 
a given function round a prescribed contour. We use, 
in the first case, a rectangle and in the second a quadrant 
of a circle. 

Example 1. Prove that f e~ x% cos 2ax dx . \ ^ire-** by 

J o 

integrating -* round the rectangle whose vertices are 0, R, 
~ , ia. 


C B 

Fio. 10. 

Let A be (1?, 0) and O be (0, a) in the Argand diagram. 
On OA, z = x ; on AB 9 z = R+iy ; on BG 9 z = x+ia ; 
and on 0(7, z = iy. Now e"** has no poles within or on this 
contour and so, by Cauchy's theorem, 

[ R f* f f 

'o JQ JR Ja 



p /j 

e~ x *dx e a * 
Jo 'o 


2oa? t sin 
t-ij a &'dy=*0. . (1 

i f e-^a-zfly+if 1 ^ <e-^ $ . e a *. a 

and so this integral -> as R > oo. On using the result that 

I e~ x *dx = i\/7r, 
we find, on making !?-> oo and equating real parts in (1), 

1 cos 






Example 2. By integrating e^'z - 1 round a quadrant of 
a circle of radius R, prove that, if 0<a< 1, 

_1 cos 

The contour required is drawn in fig 



FIG. 17. 

Since the origin is a branch-point for the function z*- 1 , 
we enclose it in a quadrant of a small circle y of radius p. 
We integrate round the contour in the sense indicated by 
the arrows. On y, z = peiO and we get 


<p a J dO = 
since | e~P sin e \ < 1 when p is small. It follows that 

/B->0, i/a>0. 

If a< 1, | z a ~ l \ < when J? is large enough, and so by the 
same argument as was used in proving Jordan's lemma 
( 46) we have 


as #->oo, if a< L 

Hence, if 0<o< 1, we get, on making p-> and 




since there are no poles inside the contour. Hence 

r 00 r 00 / TTQ ira\ 

x*~ 1 (cos x+i sin x)dx = e~*y*~ l (cos +i sin I dy. 
JQ J \ 2 27 

Since f tr*y*- l dy = r'(a),* on equating real and imaginary 

parts, the required results follow. 

49. Expansion of a Meromorphic Function 

Let f(z) be a function whose only singularities, except 
at infinity, are simple poles at the points z = a l9 z = a a , 
z = a 8 , ... ; and suppose that 

Suppose also that we know the residues at these poles : 
let them be b l9 6 2 , 6 3 , .... Consider a sequence of closed 
contours, either circles or squares, C lt (7 2 , <7 3 , ..., such 
that G n encloses a l9 a a , ..., a n but no other poles. The 
contours C n must be such that (i) the minimum distance 
R n of C n from the origin tends to infinity with n 9 (ii) the 
length L n of the contour C n is 0(R n ), (iii) on C n we must 
have /(z) = o(R n ). Condition (iii) would be satisfied if 
/(z) were bounded on the whole system of contours G n . 

When these conditions are satisfied we can prove that, 
for all values of z except the poles themselves, 

1 1 

To prove this, consider the integral 

wh^re z is a point within C n . The integrand has poles 
at the points a m with residues b m l{a m (a m z)} ; at = z 

* G.I., p. 84. 


with residue f(z)/z ; and at = with residue /(0)/z. 
In particular cases the last two residues may be zero. 

If now we can prove that J->Q as w->oo, the theorem is 
proved. Here we require the conditions laid down above 
on the contours <7 n . On making use of these, we see that 

The series is uniformly convergent in any finite region 
which does not contain any of the poles. 

As an example of this theorem we prove that 

1 oo / )n-l 

cosec z = 2z S -y-^ - . 

Consider the function /(z) = cosec z (z^0),/(0) = 0. Now 


sin z has simple zeros at the points z=n7r,(n=... 2, 1, 1, 2,...) 

and so/(z) = : will have simple poles at those points. 

z sin z 

The residue at z = HIT becomes, on writing znir = f, 

sintf+nrr)} ^ i im 

(f +nir) ain (t+nn) f-^0 (f-f nn) cos (f +nir)+ am (f -j-nw) 

nir cos WTT 

There is no singularity at z = since 
z~sinz 0(1 z I 1 } 

zamz ~ z*+0{\z\ 4 } 


Let O n be the square with corners at the points (n+ J)( 1 *)ir. 
The function 1/z is certainly bounded on these squares. To 
prove that cosec z is also bounded, consider separately the 


regions (a) y>^ 9 (b) y<-ln, (c) -fr^y^n. In (a) we 
have y>i?r and 

coseo z 

for | 6 "-a-*| > |{| a"| - | a-* |}| - |{| a-f | - | c'\} |; 

and a similar argument applies to (b), writing y = t so that 
$> JTT. For (c), let AB be the line joining the points \ir,\Tri. 

Since | sin z \ = (cosh 2 !/ cos 2 #)* ; on AB we have, since x = JTT, 
| sin z | = cosh y^l, so that | cosec z \ ^1. 

Since cosec z has period w, it is bounded on all the lines 
joining (n + $i)n and (n + $ + $i)ir. Hence cosec z is 
bounded on all the squares (?. The previous theorem 
therefore gives 

cosec z- - 

I 1 \ 

+ , 

~n7T HTT/ 

the accent indicating that the term n = is omitted from 
the summation. Since the series with n>0 and with n<0 
converge separately, we may add together the terms corre- 
sponding to n and write the expansion 

cosec z -- =* 2z 

z - n w~" 

50. Summation ol Series by the Calculus ol 

The method of contour integration can be used with 
advantage for summing series of the type 2f(n), if /(z) 
be a ineromorphic function of a fairly simple kind. 

Let be a closed contour including the points m, 
m+1, ..., ft, and suppose that /(z) has simj 
points a l9 a 2 , ..., a fc , with residues b v 
the integral 


77 COt7TZ/(z) 



Tho function IT cot TTZ has simple poles inside C at the 
points z = m, ra+1, -., w, with residue unity at each 
pole. The residues at these poles of TT cot 772: /(z) are 
accordingly /(m), /(m+1), ...,/(n). Hence, by the residue 

f f(z)7TCotirz dz = 27Ti{f(m)+f(m+I)+...+f(n) 


+6x77 cot 7ra 1 +... +6^71 cot TraJ. 

If conditions are satisfied which ensure that the 
contour integral tends to zero as w->oo, we can find the 
sum of the series Ef(ri). Suppose that /(z) is a rational 
function, none of whose zeros or poles are integers, such that 
z/(z)-> as | z |->oo. Let C be the square with corners 
(n+\)(\i). We have seen that cotTrz is bounded 

... , I f fl . , dz 7rML 

on this square and so zf(z)ir cot TTZ 
\Jo z 

for n 


large enough, where M is the upper bound of | cot TTZ | 
on (7, L is the length of C and R is the least distance of the 
origin from the contour. Since L = 8R, the integral tends 
to zero as w->oo, and so 


lim S f(m) = Trtyi cot 7ra 1 + . . . +6 fc cot 7ra k }. 

If we use 77 cosec TTZ instead of 77 cot m, we can obtain 
similarly the sums of series of the type Z( l) m f(m). 

oo I oo ( l) n 

Example. Find the sums of the series E -, E -. 

For the first, /(z) = and so z/(z)-> as | z |->oo. The 

z a +a a 

two poles of /(z) are at z = ai and the residues at these 
poles are l/2af. Hence 

00 1 I ] I "\ _ 

27 r ; SB w i r . cot Trai -; cot ( nai) \ = - coth we 

!.- m 2 +a 2 t^a* 2ai J a 

or -^ +2< 


Similarly we get, by using IT cosec irz instead of w cot nz 9 

oo / l\m I 

Z = T- . -4- r- cosech iro. 

m m2 + a 2o 2a 

In simple cases we can deal similarly with functions 
f(z) which have poles which are not simple. As an example, 
consider the series 

s i 


Here f(z) = (a+z)~ 2 has a double pole at z = a. By 
Taylor's theorem 

cot wz cot( 7rGt)4*(w2J+wct){ cosec f ( ira)}-f-..., 
and so the residue of cot irz/(z+a) 2 at = a is w cosec'Tra. 

oo l 

Hence 27 = w 2 cosec 1 wa. 


Use the method of contour integration to prove the 
following results 1 to 10 : 

, C* a dS 



. ( ,.v, ) - f (napositive 
+2 cos V5 




' J - 

cos a? 


cos ax dx ir . 



f<* a^-icfo 2w /27ra + 7r\ 

J 1+5+? = 73 cos Hi-) coseo - ' (0<0<2) ' 

++ 3 

x a dx nil a) 

/ rf z 
11. Evaluate I - taken round the ellipse whose equation 

r c?2J 
is x % xy+y* +x+y = 0. Evaluate similarly - 1 round 

J 1+2 

the ellipse 

12. Show that the function }(z) = z/(ae" iM ) has simple 
poles at the points z = i loga+27rn, (n = 0, 1, 2, ...) ; 
and by integrating /(z) round a rectangle with corners at 
ir, .7r+in prove that, if a>l, 

fir xsinxdx n , 1+a 

I - - log - 
J 1+a 2 2a cos a; a a 

13. By taking as contour a square whose corners are 
N, N+2Ni, where N is an integer, and making N-+CQ, 
prove that 

f 00 dx 10 

I - = log 2. 

Jo (l+# 2 ) cosh (jTTtf) 6 

14. By integrating e-*^* 1 " 1 round a sector, of radius R 9 
bounded by the lines arg z = 0, arg z = a< JTT (indented at 0), 
prove that, if fc>0, n> 0, 

r cos cos 

J - 

r> x*dx 7T\/3 

15. Prove that P -3 r = r- 
J o a^ l o 

16. Prove that ____&- _(J-a+l-ar-), (o>0). 
*a*+x* 2o* 


17. By integrating e a< /(e~ 2<1 1) round a suitable contour, 
prove that 

I : L. = ITJ. coth fact 

J e*y-l * * 2a 

18. Prove that sees = 47727(--l) n (2n + l)/{(2n-fl)*77 a --42*}. 


19. Prove that, if 77<o<77, 

sin 02 2 *J n sin na 

~: = - 27 ( l) n - , 

sin 772 77 n i 2' n* 

cos 02 1 22 cos no 

__ == -| 2i ( 1) 

sm 772 772 77 na ,i z*n* 

20. Prove that 

* 1 __ 77 sinh(77aV2)+sin (77a\/2) 
n--ao nT+a* a 3 V 2 cosh (770 V 2 )~ 

oo / !\n 

and find 27 ~ ^ . 

(ii) Prove that 

00 1 1 77 

27 - - = - i ; (coth 77a+cot na). 
nl w o 2a 4 4a 3 

77 sin az , . _ _ 

21. By integrating - - round a suitable contour, 

2 3 Sin 772 

prove that 

1_I + I_I+ ^l 8 
P 3 3 5 8 7 3 32' 

f 2+1 

22. (i) Prove that 2 2 log - d2, taken round the circle 

J 2 1 

2 | = 2, has the value 477^/3. 

f lg x dx 77 

(u) Prove that ~ - ~- = -- , using as contour a 
J (l+ar) a 4 

arge semicircle in the upper half-plane indented at the 


1. If u> = tt+tt> is a regular function of z, show that 

/ d* d 2 \ 

I + 1 w = is equivalent to 

Hence show that 

iv\'(z+z) 9 !.(-)} #)-#*) +c f 

\& 41 J 

where C is an arbitrary constant.* 

Use the above relations to find the regular function / (z) 
for which u is (i) log (x* +t/ 2 ), (ii) x 2 y 2 +4txy and for which v is 

e 2x (y cos 2y+x sin 2y). 

2. If x = r cos 0, y = r sin change the independent 

/ a 2 d* \ 
variables in I + 1 <f> = 0, to r, 8. 

If u (x 2 +y 2 )*/(x*y*) find the function <f>(u) which 
satisfies v a < = 0. Find also the regular function / (z) of 
which <f>(u) is the real part. 

3. Show that, if y +0, there are two points unaltered by 
the transformation 

unless (8 a) 2 +4y = 0, in which case there is only one such 
point. Show that, if z = I is this point, 

-L L+ 

,_! s-1 +" 

where K is a constant. 

* This result was communicated to me by Prof. A. Oppenheim- 


Show that z = 1 is the only fixed point of the trans- 

l+iz tan A 


l+i tan A* 

(0<A<j7r), and that this transformation maps the inside of 
| z | = 1 on the inside of | w \ = 1. 

Sketch the curve in the w-plane which corresponds to the 
straight line joining 2 = 1 and z = t. 

4. Prove that, if fc>0, w = tan (irz/lk) maps lines parallel 
to the axes in the z-plane on systems of coaxal circles in the 

Find what corresponds to the infinite strip t = & and 
indicate in a figure the region of the w-plane corresponding to 
the square Q^x^k, k^y^2k. 

5. Show that, if a>0, the relation u> = aicot \z maps 
the semi-infinite strip 0^a5^2?r, t/^0 on a half -plane cut from 
u = a to w = oo . 

Two circles, with real limiting points (a, 0), are drawn 
in the cut w-plane with centres (2ka, 0), (to, 0), where Jfe>l. 
Show that the space between these circles is mapped on the 
interior of a rectangle in the s-plane whose area is 

(2k-l)(k + I) 
" g 

6. Express the transformation 

in the form - = k\ - - 1 and hence show that the inside 
w ft \z b! 

of | z | 1 is mapped on the whole w-plane cut along a segment 
of the real axis. 

Illustrate by a diagram what corresponds in the u;-plane 
to that part of the circle | z i \ = <\/2 which lies in the fourth 

7. If a, 6, c, d are real constants, some of which may be 
zero, and 

*= z+d ' 


show that there are two values of w, each of which corresponds 
to a pair of equal values of z ; and that these values of w 
can only be equal if the transformation is bilinear. 
Discuss the transformation 




in this way, and find the boundaries in the z-plane which 
correspond to| w \ = 1. 

Show in a diagram the regions of the z-plane corresponding 
to |10|<1. 

8. Iff (z) is regular within and on the circle | z \ = R and 
^ If ( z ) I <M on ^ e circle, prove that, if | z \ = r<R 9 


where a is the constant term in / (z) = Z a^s*. 


If R = 1, a = 1 and \f (z) \<k on \z\ = I, prove that 
/ (z) does not vanish within the circle | z \ = 1/(1 +&). 
9. Show that, if 0<a<7r and 

4az cot a 

then w f (z) gives a conformal transformation when z lies 
in any finite region excluding the points z = t, z = cot a, 
z = tan a. 

Show that the boundary of the semicircle | z \ = 1, Rz>0 
corresponds to an arc of a circle in the w-plane subtending 
an angle 4a at the centre. 

Find the two points in the z-plane corresponding to the 
centre of this circle. 

10. Show that, if 

u? = 

two finite points of the z-plane are mapped on every finite 
point of the to-plane, except the origin and w = 1, and 
explain why the mapping ceases to be conformal at the 
point z = 1. 

Show, in a diagram, the two domains of the z-plane which 
are mapped on the semi -circular domain | w |<1, Iw>0. 


oo oo 

11. Iff(z) = 2a n z n and <f>(z) = b n z n are both regular within 

a domain D which includes the circle | z \ = 1, prove that 

o * o 

Hence, or otherwise, prove that 

<W- " (IT) 5 + dV- " " s; J7 COS (2 8in e)de - 

12. A function <(z) is regular over the whole z-plane, 
except at z = and at z = oo, and, for all values of z 9 
<f>(z) z<l>(pz), where |)5|<1. Find the Laurent expansion 
of (f>(z), given that the constant term in the expansion is unity. 

Show that <() =0. 

13. ABCD is a square whose vertices are 0, t, 1 +i, 1 ; 
F t is the line BG and F 2 consists of the other three sides of the 
square. Iff (z) = 2 5 +z 2 + 1 prove that (i)/ (z) does not assume 
any real non-negative value on JTj (ii) R/ (z) >0 at all points of 
r a except B. 

Hence, or otherwise, evaluate 


and deduce that /(z) has just one zero inside the square 

14. By integrating e ij>f /cosh a z along the lines I(z) = 0, 
I(z) = TT prove that 

COS px irp 

cosh 2 x 2 sinh 


15. By integrating 

cosec z 

round a square whose vertices are (n + J)(7r^) prove that 

cosec t 2et * ( l) n 

" /i_iw 


16. By discussing 

L (-7r<a<7r) 

cos TTZ z 

round the circle | z \ = r, where r is an integer, show that 
cos q = 2 (-l)f+l(r-}) CO s(f--})a 

17. By integrating z~* sec \itz sech \TTZ round the square 
Rz = 2N, Iz 2N, where N is a positive integer, and 
making N-+CQ prove that 

sech sech - sech 5 

16 35 I 55 

18. By considering 

coth TTZ cot TTZ , 

taken round the square a;= (N+$), y == (^+|) where 
W is a large positive integer, prove that 

^r 77T 8 



The numbers refer to the pages 

Aerofoil, 76 

Algebra, the fundamental 

theorem of, 113 
Analytic continuation, 103-107 
Analytic function, 105 
Argand diagram, 6 
Argument, 6; principle of the, 108 

Bilinear transformation. (See 

Biuniform mapping, 33 ; by 

Mobius* transformations, 46 
Boundary point, 8 
Bounded set of points, 8 
Branch of function, 26 
Branch -point, 28 

Calculus of residues, 114-137 

Cardioid, 62, 77, 81 

Cauchy's theorem, 89-92 ; 
integral, 93 ; residue 
theorem, 116 

Cauchy-Riemann equations, 12, 
30, 36, 89 

Circle of convergence, 1 7 

Circular crescent, 53, 68 ; sector, 
transformations of, 69 

Coaxal circles, 46-48 

Complex integration (see Con- 
tour integral), 85 

Complex numbers, defined, 1 ; 
modulus of, 2 ; argument 
of, 6 ; real and imaginary 
parts of, 1 ; geometrical 
representation, 6 ; ab- 
breviated notation for, 3 

Confocal conies, 71, 74, 81, 82 

Conformal transformations, 

definition of, 32 ; tables of 
special, 78, 79 

Conjugate, complex numbers, 
4 ; functions, 14 

Connex, 9 

Continuity, 9 

Contour, 86 ; integral, definition 
of, 88; integration, 117, 
128 etseq. 

Critical points ol trans- 
formations, 37, 45, 65 

Cross-cut, 93 

Cross-ratio, 49 

Curve, Jordan, 8 ; Jordan curve 
theorem, 9 

Cut plane, 27, 69, 70, 73, 126 

Definite integrals, evaluation of, 

Determinant of transformation, 


Differentiability, 10-11 
Domain, 9 

Element of an analytic function, 


Elementary functions, 17-25 
Ellipse, 72, 74 
Equation, roots of, 1, 31, 109, 


Equiangular spiral, 31, 82 
Expansion, of function in a 

power series, 18, 95 ; of 

meromorphic functions, 

131 ; Laurent's, 97 
Exponential function, 20, 31 

Function, analytic, 11, 105; 
hyperbolic, 23 ; holo- 
morphic, 11; integral, 21; 
logarithmic, 24 ; many- 
valued, 26 ; meromorphic, 
107, 131 ; rational, 20, 
103; regular, 11 

Gamma function, 130 
Goursat's lemma, 91 
Green'* theorem, 89 



Harmonic functions, 37, 111 
Hyperbolic functions, 23 

Imaginary part, 1 

Infinite integrals, evaluation of, 

119-130; principal value 

of, 120 

Infinity, point at, 9, 102 
Integral functions, 21 
Interior point, 8 
Invariance of cross-ratio, 49 
Inversion, 43 ; transformation, 

Isogonal transformation, 32 

Jordan curve, 8 
Jordan's lemma, 122 

Laplace's equation, 15, 30, 54 
Laurent expansion, theorem, 97, 


Lima9on, 61 
Limit point, 8 
Liouville's theorem, 96 
Logarithm, 21 
Logarithmic function, 24 

Magnification, 34, 39, 41 
Many-valued functions, 26 
Mapping, 32 
Maximum modulus, principle of, 


Meromorphic function, 107, 131 
Milne -Thomson's construction, 

M6bius' transformations, 40-56 ; 

some special, 51-54 
Modulus, 2 
de Moivre's theorem, 6 

Neighbourhood, 7 
One-one correspondence, 33 

Parabola, 62, 64 
Point at infinity, 9, 102 
Poisson's formula, 111 
Pole, 99, 107 
Power series, 17-25 
Principal part at pole, 99 

Principal value, of arg z, 6 ; 
of log z, 25 ; of infinite 
integral, 122 ; of f*, 25 

Radius of convergence of power 

series, 17 

Rational functions, 20, 103 
Real part, 1 

Rectangular hyperbolas, 61 
Rectifiable curve, 86 
Regular function, 1 1 
Residue, 99; theorem, 114; 

at infinity, 114 
Riemann surface, 27, 59-60; 

63, 71, 73, 75 

Roots of equations, 1,31,109, 113 
Rouche"'s theorem, 108 

Schwarz-Christoffel transforma- 
tion, 80 

Sets of points, in Argand 
diagram, 7-9 ; bounded, 8 ; 
closed and open, 8 

Singularity, 11, 99; essential, 
99; isolated, 99; non- 
isolated, 101 
of analytic function, 105 

Spiral, equiangular, 31, 82 

Successive transformations, 58, 

Summation of series, 133 et seq. 

Taylor's theorem, 19, 96-96; 

series, 101, 112 

Term-by-term differentiation, 19 
Transformations, conformal, 32 

et seq. ; isogonal, 32 ; 

special, 57-83 ; tables of, 

78, 79 
Trigonometric functions, 23 

Uniform continuity, 10, 87 
Upper bound for contour 
integral, 88 

Vectorial representation of com- 
plex numbers, 6 

Weierstrass' definition of ana- 
lytic function, 105 

Zero, 98, 107